Arithmetica, cum demonstrationibus Jacobi Fabri Stapulensis. usica libris demonstrata quattuor, Epitome libros arithmeticos diui Seuerini Boetii, Rithmimachie ludus q t pugna nueros appellat

Iln boc opere cortrcnra. aritbmcaca occem librio Dcmonftrata jf^uffa libno ocmonftrau quamio: epitome ibb:o0antbmcncoooiui @cacriru X^octq l^tbnumacbic iuoao 4 1 pugna nucro? appeUaf e.eontcnuo CabilonenlTdrm lauDc Sntbmcacce-r jfeuficco. ICempote iam multo ooac latuere fototcdt duae retinet comitee Haua minerua fuao. t^ucplaaoc terrae poli tempora multa reuiTunt# 6rata quoc^ ante alias £aRja terra placet. bie clim cdd>u6 futt omms acaica tcUus. tbagoza patnam oIlTugicnre famon. f^cllaoa nunc linquunn et ooctas paOjDls vrbes* 0equanior9peomt/parbtfiorc? lares, bccveniiomnunooanumcronim einaaeatcTtia* 3t<^ Docet numens qumquio in orbe fitum dl. altera Dularono cantu/Hoiburcp canona £oomuiRcviros traDiiur at(s fens: ducrua'Picn|s tenet vmu nomina mulla Clf nicbil hac mulis gratius dTc Tolct. Sttamen amiiccmibpulcimrcrc manne: dui leta boc Ihioiis fronte Dicaret opus, bocfolum ihiDium ate^ bcctUi curaumiarc/ ^Irritus et ne ili Dirpercarcp labor. THoiia pmfntatio m^ojDan» per obum fabnT It.ipu!cfem labo 00 danlTtmnnj virum 5o3niicm oc^aiiay p*cfiociitcm pariilciifcin. Suas mcPignc3monuilW£lariflimcvir:quot cognita pir.oDircitcoaooucatPifcipU pnarum parcnoaritbnictica/^tcB ignota relinquat tcncbuo.vt que niiUiuo oifcipUnc fugiatlhiDium.^ri paimioctiilcgumautotitaoan quibuopoiiifimuiuavfac opera; fibi nuniciie egere vifa cft.vt q«ifticu vnicuicp fi oignuatc oirpcfaimunc Srubmctica/mlc vot6conictncar6ncpftantc.quc3groc/agro:u liimtce/aqua^omcrgia/alluuioitcfcDtpk^ racg fifia foUicitc pfiocrat q4fiaibcniatico ocRitma pltoio imii^ plane alTcqui valeat.ll^inc iinpcratojco rboinamBcfparian'’'aDnanu0'1Crai3n'’/Sbcoooriue/3Crcbaoi'’ylbonoauo/ Ccnftarinustaliop^plurtmi legatur pcritooagnmcf02Mmlhtuiirc:quipoDirmo2um rocni ao publicam vtilitatc tenerent qua nuiiicroppficitfagflcitas.lblcilucme jboocratuoilU:? TCfi f^oiofuo mfignem libcllii rcliqmt.lEt 2ritbmeiica cotepUtio rite cognita facilem pbet muficapmooulariotiaintcUigcniia aftrcnomicariiepfnbtilitatu mgreiruo.ctpafcoSbcolo: gia iiumcno olim vt quibufoa qd omina graoibuo tota miicbafiqmio tnue in facrio litterio fua retineat myllcria nucri.CoHcigif numeroo/numerotuep DirciplinailcgcoiniBficto.iuj ftieia ccca rclinqiur.nulla mooulationu repenef rcgfa.tiulluoctlclim ptcmplotionu noitao. facra? Ifapocluebut myftcrianme zvmucrfa pbilofopbia qpanicr buinano?/ oiumotiiqj cognmo Dcfcribif.«Sucquioc pmooitateo aiTccu te oifcipline/atip mcomooa neglecte pfeoo mime lamcrcivi dImud olio tramite trcoiocrini $pea inccoctcs noo poffc (iccp Digna fatio ptcplanbceognorce.iby tbagotaocni fine ruero? pfioio nicbil poRcrciri ?tcocbat.crif>Iato mfuc acaocmicvcftibulo bocinrcuiplitcpigrima.flniiobuc matbcmaticc expers inirocar. qui itoto ferme £imco oc nafa re? pnumeros oilputa t:z in octauo znono rctpublicc mfta oc bac re oincri t.quc SCbcon Tmirncus matbematteus ob rci aromtatem intacta reliquit. ^uap:cF^ non ab re oolcbas bac numerofam bm^olmi parificfi (luci) pbilofopbaiiu turba et bona? lfa?cupioa:ra iiecciTana fcnitta. cum ao omina aiiurgeoi lumocfccocoi ao buana cfrcDcIlitutam.aeapccrgomlebilfactotvmccoceemaritbniciiccoifciplinc lioioani cla^ riiTiini viri libellos nona ccmcnratioio Itrccnfis labonbus iliuilratos/ ttiocp nomini Dicatos qui eomoDitatcseimmcratas kcii atferre valcbut fulfjciefiiriinc.iSt fi mbcoopus tuisaufpi» cqsabrolunimiamniigrct aoccfcros.vtvclficinrclligani omnes picferiim nbi fc gratias babituros per quem fuerint banc litterariam facultatem affcciiii.Bale felicifiimc. 3Irgumciitum occem libtotum ^otoani. •Jbrimus palTioncs numero? comuncsifua? partiiim/ct quatfi e? oiuili numeri partibus fiat Direutir. Sccunous cil oc p:opo2ti’onum et p:opo2ti’onaUtatum comunibus partionibus. Certius oc numero ptimo/compoftio/ao alterum ptimotefnumcris maliqua p:opo:tiotic mintmis. ^uarrusoc numeris continue pJopcitionalibus/comcnfHrabilibus tmcomcnAirabilibus. Quintus oc aooitionofubtractionc/ct patiinone ptopoinonum. 0cxtus oc numeris quaDratis/cubids/rupcrlicialibuo fimilibus/etfoliois. Septimus oc numero pari/impari/paritcr pari/paritcr ipari/impantcr pan7pcrfcctis/abu= oaniibusietoiminuns. t0ctaimoDcfMmismimcro2um:trigom'0/rcrragonis/pcnfagoni6/bcxagonis/bepragoni6/ oetogonis/pyramioibus rcrratilibus/cttcffcns. lHonuooccqualitatc/mequalitaic/miiIcipIicibu6/fupcrparticularibus/ct fiipcrparticntibus Dccimnsocmtoictatc :$riibmctica/Scomctrica/4(buf[ca:ciocmcoictatibus minus piin^ cipalibiis. C'5o:D3ni 'Bcmonry Oanirimiviri dementa Sritbmettca:cu pcmoftrauoibiis 5acobi ^ab:i StapulcnfiDtao 'Joanne Dc t5anaYSciiato:c|i>anficnfcm» fliiae cft rci per Te oifcrcno.lBumcrus cft quatitae oifcrctotum coU Icctiua.TRaturaUo fenee numerotum oiaiurun qua fm vmtatisaos Viccnoncm fit ipfor compuiatio,©itfcretia numerozutnappcUaf ille: quo maio: fuper miozcm abuoai. ‘flumen ab alus equioiftarc oicu* tur:eum ipfo?: ao illoe cqualcefuntoiffcrcntie. flumemo per alium multiplicatur:quiro«cecoaccruaturfibi:quotice in multiplicate cft vmtae.ct qui ej: multiplicatione concrcfcir.pzoouctue nominatur, ‘ftumerue alium numerare oicttunqui fm aliquem mulfiplicatue lUum p:oDucit.t^ar0eH numerue: numeri minoz maiozieicum mmo: maiozem numeraut qm numeratur: numerane multiplc)cappcllatur.©cnominano cft iiumcrueifm quem fumitur pars tn fuo low Similco Dicuiur partcs:quc ab cooemnumero ocnommantur.flumcrus fm quem aliue DiuiDimnoiuifoz nominaiur.partcovero in quas DiftnbuituriDiuiDetia appeU lantur.-^zima zfimpla nucn parsieft vmtae.quaoo duo numeri pariem babucrtnt c6cm:quotice caocm pare fuerit mmino:e:iot partes maioris Dicetur effe minoz ci tote partes quoties ea comunis pars fuerit in maiozc. C^ignitaics. ^Ibnis numeri pars: cft mino? fuo toto. 0.afenis minoz cft:que maiozem babet Dcnominaooncm ^uicunc^ cqualium (luceiufoem eque mulripUccs fuerint: ipflquoc^ cruni cquales. luibus (Dem nuerus eque multiplct: fuent:(Tue quo? multiplices cqles fuerint: tpi etiam funt cquales lOmnis numeri pars eft vnitastab ipfo Denominata. Cuilibet numerustotus cft ab vniratctquota pars ipflus eft vniras. Bivnitas in aliquem numerumoucaturtiiuc iocm mvmtatcmtfopfum pzoDuot» ^Ftrcmozum Differentia ce Differemus cezuocm an mcouim cft compoficj. 01 numerus numerum fupcrcrioid^cnnammoziaDDitataut amatoze fublatatnui nicn relinquuntur cquales ciDcm cquamur.intcr fc funt cquales. dfiab cqualibus cquales aut locm comunis DcmatunrcUnquiitur cquales. fi ab equalibus Dcmpti fmt incgualc6:rclmqucntur inequales. 01 DUO maiozes fimul aoDdtur pariter tduo minozes ftmuU maiozum compofitum compoUto mmozum maius cuaoit. 0i equales cqualibus aDoasitoti quoep ficni cquales. £:t fi inequales cqualibus:toti erum inequales. iPartcs fimuUfuo toti cquamur. ^uozucuncp aD eunocm numerum pzopoztio eft vna:ipi inter fc fum equales. £»uoties numerus anumero fubftrabi poteftitoties mcoDcm numerabilis eft Tp>zopoznones que ce cqualibus conftantjjpoztionibusimtcr fc funt equales. ^uo?DimiDia funt cqualiaict toti funt equales. dP>ctitioncs. tElilibct numero:quoilibci pofle fumi equales. eQuolibet numeroialiquem ^tuUbet effe maiozem. 0cricm numerozumm infinitum polTccetcnoi. flullum numerum in infinitum Dccrcfccrc. ^fiaius non numerare minus 01 numerum oamm numerus mulriplicct: iDccp pDUctum oiuiDat: numeru Datum rcDirc.ee contra It aliqs numerus Datum numerum DiuiDat:? quoo pzoucnit rurfum multiplicct:uiDem numerumoatum reotre. C^i9nit3tc6at(7petltionc8pduc38vtt>tfc4>UnarefaciUo:eTbi(9pic(iet;aoicc(nm8:au38pafnm neccirartao can^ omnlbue notas autotfupptelBt. 11^ IiS €^>mtiw riutnmicnnnoi:maio:f0 aut paro cft aut partco. , CSintaJl/c/oquotUbctnutnm-ainaionctbcsmfnotce.oicobcfrcparfcm niitetceawcfonn- TBaHiau[minoinuincracmaio:nii vto numerat a:cttuiico per PiffimrioticmcUparea.autniioz no nunicratmaioteniqucaemooumbnon numerat arfcseUaliuo cos eoltcrmimcr.iiisquirirc.cctmic quotus cfuerit in bmino:c:tot partes bmmozpcr vltimam oilfinitioncni ertt amjiotis.qij Umto: netinomeraimaiotem wcmon numerat a.nccpnumcruo cois eos numcrancum per quintam concc= ptionenianuftibetnuinenwiitaefitparsabipfooeitomfatatpcrvltimam oiffmiiionc3 iocm elficts. conUaci^turocmmiiioicmniaiotiscircpanetn autparrcs> CiOmmo iiuincruc’’drcum fc poritc2um zcqualitcr ab co eiftaiirium cll mcsictao. t conbcm fucnimcoictaoailooab co cquiDtftarcconucnict. C‘-5u aquifuiii® numerus: ei bctecirnimpofinct abeo equtoiftantes.bmaiot 5et cininot.et ftto Piffcrcntiacomumscioembaoatctaaoc.firipenumcruscdpofitusepbtc.oicoaenc nuoietatcm c.etliaficmeoictaseisieobctceircumporieosequioinareab a.T^umumaiitenific oftenDitur.nam 5’iumafuperaic:tmbfupcrata.Dempta crsoocoioifferentia ab btperfcbam partem noneoigntea tisrcfwuubequatura.feo etcaocmoiffcretitiaoaDoitacqiSfitfperptimam partem ciuroe otgnu tatis equatura.crgopcrocamu^tcqumrcriouQbctfqueaoem numero aequatur; tnfrecquabfi! tur.rco refibuumbet 0et cftfeoecimarettaoiffnitafcm equitur e.igifur 7reflouSbt fqueeqiiaiiE rer»Biiobefbetcfuiiulcttacquature.etrerit>uub!7fmonltr3tafuntcquaUa.lgiturfcft meoiccasc f<t)fmoii(lratuseflcquari4:isituramft>ieta8cfte.qOenpi[mum.etmon5usaU)B cofimili agatur arguinento.SciSmautricpjriitameoietasnumcropbtcfimuUunceop.ribcnoncquiDilfantiiucro atcoBOifferentieacaperoiffmiiionemnSruntcquales.ritigituroolffcrciitiaaaociIJoBnumerott flimimuina9rooaoa:7eomporirusfitg.qugimraaocpi(ferettacno7eaDcmolfferetiacflsat.a iguurpcroiffimtionemgtcequibtftatnCeroa.qjcrgog tcequiPiftattta pcriinmeowte mollraifi aipfo?fimiiliunctopeftinetMctas.fct>7aponifmeoictasbTeflmuliunctop.pcr tcrtiiigifpteptis onembefiinuUctgcfiiTiulciBcnumCToaequcmuItiplifcsaoinuicemfiintcquales.abwocpigiiur tCTipto COI numero eper vnoeama cocm fcicntiam rcfiouib et gaomuiccm fuiit equalcs.quciaitur oiffcreniiasaoaieabcmcritbaPa.quarccquipiftabutb tcab numero a.qij cll ?rraJ?vpoil?eiimer <LSt 0110 iiumcn’ aouobiio nuiiicriocircum repofTtio cqualitcr offtcnr: illio pjuctts ^ erunt cqualce.^;^ «o cqnalcofucrintMb ipfis cquioiflarc ncccITcclT. CCiaocm quelnp:cveocntipem5Hratioeft.Sintigifactbintcrcctocquioillate6:cmat(muBet o minimus ctftnffcreciacoisnitqseppofltuB CSC 9:ctrfiaca5 afit g.oieo numeros nbJimuUunetos equaleselTecofimuliucris.eiptraneolTmununcttfuntequalcsabrimuliuctiotabequioiftantes el)eco.T|b:imuparet:nam caooiffcr^tia fitcao efiflemjo numerus quo fupcratenumcrCc-gpria cstremoveaeoflsbitesefoifFcrerijsfcjestrcmopaDmcoiumc.fillciuvtoftenfumcrtooiffcretiae abcetfPifferctiacaoaperijTPotMimtcflpofitaficfcoisoilfcretideaoactbaoo.aeq: bcotlates eifbem;qi5 patet nameu ffitmfferctiabao oicafiquiPeoiffcrctiafaooitamlnotl numero bp nona pccptwemrco5itb.ellitaqjbcqu3li8g.qnarcepftabitcS8b:cepercloe3pcepitonepmru3tureec aetgfuaqmoemolffcrentiaabmqueequatunprib.Suncigitursbifictietcomncti ciocm tertio cequalc8!aoinnicemcquales.q6eltptimum. Burfumoicoficofimulcquanturabfimuberiaab cquioinarcco.-fflamcapioljqufequlolfletaocg-iambaop:quil?bquianc cqiumfiabuntiocrim* tncotatcmonnr9tuml?bfimuUqttaturco.feoet3bponururcift>cmeoftmuUunctiB cquari-quare l?bct ab ct?cmrertiocqualcs:aomutcnnequab£irur.Subftraeto Igitur ab vtroi^ <olb:reinnnicntD ccemfciam fj«3aoinulcemequalcs.crgo equalismfferctiaaaotetbaoo.quare ooifftmtioncin aDtnuiccm cquisiflabunt.quoo ciltciSm aicp totum ptopofitum. CSi fuerit putniie fetf iquota pare tertius qiiartucrunt primus ct tertius tota pars rccunoizquarti:quota pzimusfccunoi. CSitapiimusnumcru6:bfccunt)U8:ctcrtiuotet5quarm8.qufaatorjparseft tpfInsb.-qiiotaBB ccflfpfius otintclligo bcr 0 mfimilcs illas partes oiuifos.ct quia prima vnius cum prima alterius cll tanq>a ctetfimtliterfcSacumfecuca.ctbcccomactio toties fieri poteft quoties pmueInrccOoo. toties igitur numerus equalis acptimo ct renie:fumi potcriin bofecfioo iquartotquoties ain b. 14 iquarcacfimulpumuscttertiuopcroitfinittoncmcrOtflmiris et tota pars fecuiibiet quarti quota ptimno fccunol.quoo cllptoporitum. eSifuerit primus tote partes fccunoi quote tertius quarrirerunt pziimis ct tertius 5 tote partes fccunoi ct quarti:quc ic ptimus fccunoi. C£5itvtpriuB primus immcrusnirecuousbriertuisciecqiiartuoo.ntqsapartesbfumpte fimeet I—bencminaicinbfmf.fitiBgvtnaillanpamumaietljwaparciume.cuigitiirgptimustoraparsfit ysY r'l’t«“®“l“o«t’'ftiuscn!jqu.irti:perp:cmi(T3mcompofitumcsgctbpnmo7tertiotota Ds erit ’I»>bsrecfitnctqiiartiiariprcitoininarafmf.arpartcaactcrunmturfiueieruntigiturpartcsactcao 00bDfumptcfmeictoencminatefmf.quarcctfinulcsetquotcpartcofuerirainb.qSdlptopofuum <ESi fuerint quotlibct numeri rotiocm aliis eque mRiplices: erit quocB co tiipoUriio < cj: CIO coiiipofito cpillio eque mulnpicy. I Ceintfl.b.c.p.c.fquintibctnunifrunntcp&efeq inuUtpllfceabcfmgulisrmsulirintdUgouatp apJitiiiitit.b Iccm.b fcrnmtt.cquflrtiimrcrucppfr pfnultimamcomporituu «i a bpnio-r terno tou pars cotiipoiiti cjr cefccSoo t(Quarto quota pare acft o.quarc compofituo er oceque miiltipirr a:' compofitum et ab vt oao a.5^acio Iratp numerum ct ab compofttum p.nmurn:» compoftlum ec oc ffPm er c ternum cef quartfirercofimilirer argumcntor.quU quota pars abae oc:tota para cao f. copofitiie igirur cy abicptimo -r tertio tota para erit compotiri ct ocet ffecuoo ct quartorquota paroabao tie.qre<ompontua ct ocfcqucnmltiplcE aocompofituabcvtoaoa.ctitaficoplurea nficri fucercfeercrifemc Ttarcpteceocntium copofito tftp oum abvlrimfi recubaaicfftctcfcs ^jpotiru. 7cSi quottcp vnifao tn pjimo rones fccuoue in tcrtiotquotics vnjtje jn rc5o:ronc5 pnmusin temo. Csint 3bctrea numeriia piimua-b fccfioua.c terttua:ftt<5 vt quoriea fuerit vnitaa in a. toties fit bmc.Pico ergo q> quotico vnitaa fuerit in bitotica acfle lit c.otuioo cni aptimum nfierfi tnvnuarca fctertium in rorit>empartea:quapqucUbefmtequaliabrccunoo.qut8eni5quo nes pnmaparaa in prima part ectotica feciioa tn fecGoaict tertia Itt ternaier ita ceincepa.ergo pet ptcceoentemcd: pofima atota pare erit compofiti c:quora para pztma para ciue ao ptimam parteme;ct cequemul ripley ao avt eiuapnma para as pzimam ptem a.at quoriea vntraa in bnoriea prima para ain pnia parte c.Wam vnitaa equaturpmc parti a:t bprimeparti r.ergo quoilcevnitaa mbfecCooriottea a priiuue inc tertio.quoo cratoeiTionftranoum. s€Si slccrna fiat ouor tmmcromm muInpUcattofiocm numerus vtrobicp ^ucnict. Cyrfi 3multiplicat bct pzoucmat cioico ctia fi bmultiplicat aiiiPcm pjouaiirc c.nam Ii amulti» plicat bec ^jucniat c:pcr ciifmitioncm quottee eritvnitaa in anoties ent bmc.ergo per pzemilTam quotieevtiitaa tu bfccuooitotica aptmua in cfertio.pcr oiffinitioncm igitur fia fibi totica cocenia tur quoticavmtaa in b:per oiffinitionem aper bmultipUcarur.at cum l?oc fit piouentt c.namo^barii cii atotica coaceruari in ciquotlee vmtaa inb.confia tergo piopofirum. fCl iQuod fit c:c ouciu alicuius numeri in quotlibctitm cit $tum c(l quoo fit cf: oucm ciiirocm in compofitum cj: illis. C^rianiimeruaquiPucscurinbetptoueniatoietincetptoueniatezoicoefgo^eompofitasejeo e^ducitur ey Mtetu ain compofitum bc.cfi enim bmulriplicet o febm a:b per otlttnicloiiem nilcrat brcUmatctperlDemcnumeratercSma.runtiglniroet eeque multipliceeb et aquare per fcycam compofitua eyd ceque mulripley ao compofitum cy bc.Sit igitur fpioouauaey suetu ain ppofito bc.cfi itacp bcmulriplicet ffeSm amfierabit Igitur numenia bcipm ffeSm a.qu3re eque multipley faobcvtoaob.feoeioembcaggre^tueoe^Jbatua eequemuuip(ey.funtlgiturDeetfeioembc eque multipliceetper tertiam conceptionem aomaiccm equalea.£fi igitur qS fu ey ouetu alicuius numeri inquotUbet ti!i:$rum qS fit ey ouetuetuibe; in copofirom ey ilUa quuomoou erat^ofuB. C^uoD fitc):DuciuquoiUbctnumcro2ummaliqucm:cquumcftUli quoo fitcjccos pofitoillozummcunocm. <ESit caocm efi pieccocn ti brpotbcfis fc® coucrfo ozoineivt oucatur bet cin act pmueniant oct e lOito quoo fit cy ouctub ct cin afingiUarim eqim efie ei quoo fit ct compofito b cma.nam cpieceoc rem qS fit et ouetu ain btcfingillatim:cqufi efi ei quoo fit ey ouetu aui compofitum bc.g alfoarim peroaauamqS fitctoucrubctctiifl:cquficftciqiioofttcycompofitobcmefioeni.qtf cft^pofitu, II C£Juoo Iit c]c Duciu quotlibct numcro:um mquotUbet numeros :cqualcc aquoo lif CC compofito jpfoium ni compofitum cpaliis. abcquotlibetnumeriquioucaturinoefquotlibet alioavtquilibetprimotumoucaturln qudibetfccQDosioicoquoDfitecouctuabcfinglUatimmqudibetalio^oc feqle efTeei quoo fit cy Ouetu compofiti abcin compofitum ocf.nam qS fir ey ouctua bcfingillarini tn o:cqu6 efi eiqb fit eyppofitoabcmo.ctcofimiUteroeeutriiabetcfmgillaiimincetfargumcrabcreqtf ntcbtlaUuo dl 5» qtf fit cy Ouetu abct cmoeec ffingillatimipcr picccoentcm cquale efTc ei qb fit er compofito abcin 0ect ffingiUatim.St per nonam qd fit cy compofito abcfingiUarim tn ocet fiequum efi ei quoo fit cy ppoftto abc:tn compofitum oef.quarc coftat quoo fit ct ouetu quotlibct numerowm in quotUbetnumeroa cquume(rcciquoofttcycompofitoillopmcompoftrumcyalife.£tbcquattuet ectauamonaioecimaiet vnoccimatpayim multipUcanoitp mooumiplanc oecldrant. ,, CiSiuicuntj numerus numerat totum ct octractum-.numcrat ct rcfiouum. eSitabtotum :a oetractum et brcfiouum.Otco q> fi cnumerat ab ct aripm nficrare b.numeref eni abfcdm numerum oectaquioefc6m o.quia igitur per nonam q-nimcficmocstifi eficinoetciuc fit aequale efi ei qS fit eyctn oifgitur aefi eo qS fit cy cin etequfi efi ab.quare qiJ fit cy cin e;equC efi b.fco enumeratqS fit eyeincperoiffinirioncnKigimrtbfua equale numcrabit.qfi efi ^tpofitfi. ,5 cCra cll q6 Iit er ouetu nOcn in fe :$tu qtS Iit ce ouetu emrocm In oes fuas partes. aquicfiqt nuerus que femei pono oiuifutrein oce fiiasptes 4fmt bcoett fetnel inoiuifuititc qfl fiet ey ain b y in c1in 01cBnona equfi efici qui Iit ey ain ppofltfi ey bc0c.at ppofttfi bc0cseft a.qfi fit igifey ouetunumeriin oesfuaspariesicquilcftctqfi fit cyooctuciufocj numeri in fe.qrcz al^atim per octanamqfifffcyouctnnfieriinfettmefi §>tomqS fit cy ouetu eiufoe in ocafuasBtesj aj Cflumcro mouo wuifo quoo fit cj ouctu totiuo numeri mijltc? lUo^tcH cprum qiJ ,4 fit cc ouctu ciufDcm in fc 7in reliquum» C3tt;ibnueruein9uoa‘tboiuiru8:cicoqtf Utetc^ucrusb fnAequQcireflqtffitej; ourtti aInfc ‘ilnb.ctlta(|uc<bq6litecAbtnb:cqua[cccd(ldfitei;bmfeib ma.oucattircrgoa bina:tuacq6 fieteictiumiabtnauocrauain cqu6 ericciq^ficcroucruatnab^atqb lircc^uccu aUcuiuainatitt bequacftpernonifacroet^acmincotnpjfuaab.crsoc&aoudfirT f«ipmattnb:tm fa:u qirum •bma.eticaarsumenrareriab9ucanirinb.ctconrimiUmoDocoiidu9e8 p:oporicum.'Sb8rcfq$ili clariwviDertftbieponis act bl8 ponia b. C6inumcruoinDuoci[uicanir:q6litcyDuctuMtiU9infcnncfi:$iumquoo if Duau Ttnufcp oimocnuum mfc :vniuubis in alic^* <[;SUabiiummi8m9uc)a7b9mifu8:9icoqblua;9uctuabiitreequ&cceEquob ftrcr9uctuafn l'eTbinrc:etn:9ucmbieamb.'aianiq6fi[erablnfc:equfie{ldquool!cct9ucru ctU8 ina ‘zbprri ocdmam.^tqb iitoroufru abtoriua ma?in bpp:ece9eiuan bis fumpranuequu efiet quo9 fit a: 9uctDamrcnnb«bmfcitna.rc9q6 fit ctambtblnaaUcmatimEoctauacquafciqb fitcyou (rubia ainb.quo9litiS(fecabinrc:cqu3tdquodAteraiiiretbmfc:crerabi8inbqb .pponeba tur^te;cl?occognDrdrurino9U8<rlickn9ilatueccrras9nuin» CSinumcrusmcuoomioafiqdfitcjctofomfcdlcoqdfitcp altero in fc ftricquu i<f cft ci q6 cj toto milluo locm bie:qiJcp cx reliquo in fc DUcto 4:Du«lur. dSirrtptiue abnumerem 9uoa‘tb91ulfu8:9koquo9 ;p9udfei:abtnfe(;li eoquoo Rt er ain Te: equficfledquo9fitc);abbtB ma/tjb in fe.iRatnqii fiteeablnfejeptcccucntcmtificUcfriiquoB fit ti:ainfetbinfc:erabi8tnb.quo9er3ofitej:9binfctiJeoquo»fitcp8mfc:tAcrit g>tumquo9c^ ainrebia-tainbbkf binfc.re9qu(>9fitecainrebte(etbismbp<r!4cqua£nquo9fitq: ab (na bie^ia99cqui}9fitcrbin(‘c.tunc4<9quobfitc^abinrccucoqao9ficc)eain('8:rqu9btte( quoufit dc abIn abist b infe.quoo cU proporimm. CSnmmcrueinruoDiuiDaturqdfttcEtotomfcicquucftciqd fitcjc ouctuvniuo 17 pamo malmo quarcr:cum co qd fit cc oilTcrcntia in fc. eSit abnSmie in 9uc a-r buiutfiis Tire? bmak: rlutoafcB ttep bin 9U3fc% in9 cqualc c9Fam ba» a.9ico quoo firrr abinferequu cc ei queo fir et ructu quarer ain b>cin prccrni.rquin ta quopfirecabinfercquucftci qtiosffrccabioinbtalnrctbinfc.atDpcercntequoD fitccbCfccu ain fecquali 9rnt prul bicquaf ci quor fir ct bbi8matcin feqre alrnatim Boctauam tci queo fit eca bl8in b-I cin fc.^^if abin fc tanma cft ^cue aquater inb cii rificr^tta cinTc.quoo intcii9irur. CiilD fit cj mmozc oiuicctium in fc cum co q6 fit cp toto in co:p Differentiam im cft is ^rum quoD 4)ucnii ct maio:c cotuoemper fc multiplicato. eSit9numerus riuifue inb maius 7cininu8:fitap9 9iffmntiaba9 c.cicoquobfitec cinfereum eoquo9fitcj:am9:tific>req’tumquoofitc;cbinre>lRamprr8ccimamquo9fite;eain9:cqusfei qb fitecbin9 7cin9.Sititai:piUde.ctqulab9iutfumefiinc7 99iiFerent{3m:peri4quo9 fitecbino alte^ 9lute<nrium:cquumellctquo9 fitcpbinfctm c.ctcumpcr octauam cm»ctiamcqnefsine; eritigiFain9equumipi8 9inre7bis9incntnul.quibuefia99a8cinrcquo9ritf:totumquo94)ue nitetainrtcinfefcjctfequabif oinfelclnfe etbis 9inc.atb{nfc'cumtpm biulfnm fit inrtc autcqualc) perocctmiquinratmetiaeft ^t&binfectnfenbisDincrcj^tuin fitctouctuvtriufqt partium in fe:t vnluo in alrerambi8.£(li9if quoe fltej; ainr roto fc; in otiferentiam: cum cminotc parte infer^tum quoo fit cpbmaioU paneinremultipl(oua.quoo inrcnoitpropofirio. eSinumerae per duo cqualia ouoqp incqualia fcccmr :qd ct ouctu vnius cqualiu mfc ptoDuamr iifi eff:cptum q6 fii cp Ductuvnius incqualium mrcliquu cum co q6 fit cp Differentia in oiffcrcniiatn . C^(ul5wiUamuea9 ocmonfh-ationcmmotam eijtium arbitror q-quSooali^fftmmerus vtab in ouo cqualia pariter tIn Puolneqaaliavt in aminus 7bmaius9tu(9ifnlla900 inequalia aovna equalium quoo totiue mcotetas cfireanoan at® cocm l^abcreoiffcrentiam.namcum iltorfi ouorum mcqualiumftraullartcto?lUuocquaUlItmcoi«a6:iIlaabcocqui9inarcperfecfioaj parrcnilccfioc buiue nccciTccihquare ao ipm coembabdrunt Diffcrenriam.lJlc fit etiavt oifferetla vniiie Incquatifi 8PreIiquum:oupUnta99iffcrenttaincoemipnu8cquab8aripfa.TRamcumipramcqual!avia7b etfrcmafincctiplimeqDaictnctiunirperoctaiiamconccptionem oiiferentia ipfopconftitutaelr tt oiffcrcnttfecorum ao ipm meoium-Sco b« Harfm nota funt.flloeergo ao propoflti oernollrattoem couertamus.Sit abniierus otuifus iifa maiorem portionem 7bnunoremrfitqj crnaciuBmreirtaet «oilteiffcrdiacomunlscBoact aob. flamvtoftefumcIlcaovtrunqjcanoecommunFcpbabcbit 9ifferentia.^o ly fit ct cin fe tm effe quontfi qd fit ct ain bcum eo qb er oin fc •qiem coimfum e Inoetb.iocoper9cctm3quintaq6fitetcfnfccqufi ceiqSfltejcoinfeetbtnfeetbtoobte.ctq^ maiorportlocfttorficteiu8partcebct9iffercttaoupUa9 9.ergo pcroccimaquaitaainbrantnea quantaqtl fit ctb infcctbbioin o.afoc igitur clq6fit et alnbnumerfiqui fit et oinfc:7 tunc totu qt}fitctainbetotofecqoabifeiquo9/uet 9infcbtnfcetbto®b(«rcu( 9cntonttratuni cirequan <in fc.ccnftjt igitur q6 Ut noacruvniaa equaliu In fc tanru cfTc q-tum q8 cc vno fequaltfl ia reliqui cum coqdfitetoilTcrcnnainoiffcrcnnj. CSi numerus per dius c|uanibct omilioncs Diuioamr:q6 fit cK oucium-ipimiDiuh Dcnnfi mmmimu com eo q6 fit cFOilfcrctia ipfius maximi ao altcru tneoioium oiffc rennam ciurDcm ao mimmu tantum cft ^lum qo fit exouctu ouozum mcoiop vnms malterum. C! mpzeccocci io anre anlmasucrtcre licet q’ ei quilibet numerus btiabusl^oc pacto Diuifionib^ fecatur vt avnaoiuifione in b-r eer feSa in oc.quoia bfit mapiniuB et eminimus i»mc&io? maioj et emmot que cifferctia pnrni ao fteunofi canoc cficrertU ao quartS tpfis fic otoinaris. THam qr oe equaksfuntcircupofittsbcmbtUiepcrfecunoapartetertieequioifilt.quareqoiffer^tiabaooque fit f:caoe erit eao c.cr etiam que oilTeretia bao eque fitgteaoe erit oao c. iHam per oaaul concepti onem Dttfercriacttrcntoru beconiitulturefOiffcretiieb ao oct oao e.t oitferetta oaoceirkemotfi per lOc conlhtiittur cp otfferctiis oso ctcoo c. Seo oifferCtia o ao cefi vtrup comunistet oiffcrftia eaocptc>bacacflcquali8Oiifer^icbaoD.enmti0iturDiifcr^ebaoetoaocc;ccquaUbusquibcm coiifiiturcaoiiiuiccm cquales. £fio ergo anumerus eo qui pofitus efi mooo oimfus CTOifferentic vt pofite ftin t a(risn3te.oico ergo qb fit ef ouctu bmapimi oiutocnrifi in cminimu cfi co qb fit e^ fin S:cqu2lc cfTc ct qb fit cp Oin evniuavioe[5 mcoiotfiin reliqufi. qb ent fit cp ouctu otn eper nona ao* inii.lteocraiiattn efi^nimqb fite:coincetinf.lHameoiuiruincfiincctoziamf.Seoqbfitepoiiic ct fper ocraua cquficu ct qb fit ct cct f in o.ct qb fit cp ftn o equaP per nonam ct qb fit cp fin cts*at qbfitCRfinctmcftperoctaul$tumqb epe inf.cp aut cp ctno ain ftm efi per nonam ^rumqbepc inb.confiatigiturqb fitccbinetiftelTe^cumqbepcinbc&eoqb fitepf ingaptopofttum. CStqiiom pars totus tonus tota pars Detractus Dctraai: cntrcfiouusrcfiom tota paro quota totus tonus. eSitab numerus cup^octractusfitbtrcfiouusa.ntip alius numer^co cuius oetract^fitoctrcfl Olis cTquota pars totus coefi tortus ab:rota pars fit ooetractus oetracti b.oico ergo tota partem crefiouu efie rcfiout aiquota pars cft totus cotetiua ab. capio numeru ecui’’ ctota pars fit quota 0eft b.ct intclitso quattuo: numcroscptimfi efecunoS otertiu bquartu.conftat entper quarta qzc tota pars eftequotaoeftbicompofitfico totam part?cflecopofitieb quotapars ceftcTtotapars pofitus eft coao ab.cfl igif abeque muitiplep ao covt e b ao coet per tertiS conceptione aoinuicl equales.quare eeftequalis a.^tcao etita quocp cao a.feo cao eefi vtoctracn ao oerraetS ttotius aototum.cfiigiturcaoareflouusaorefioufivttorusaototumetquotaparstowstortusquoo^ ptopofirum. CSiquote partes totus totius tote partes fuerit octracms oetracti ierit refiouus rc fioui tote partes quote totus tonus. CSit ab iiumerue cuius oetractus fit bet reftouus a. tcoalius numerus culus oetractus fit oct retirijusct quote partes muococfirotiusabitotcrit ocfract^ooctractib. tOicoergoctorasprcs eife aquote cocft ipuug ab. Tfbone numeru ccuius ctote partes fintiquote ocft btargumetarc pe» nitiis eooemoDo per quintam quomooo tn ptcceocnti ar^metatus es per quarti. f£)mmo pars parns: cft pars tonus Denominata anumero qui fit ex oucm ouoju nmnero? vmus maltcru partes lUas ocnommaiuiu. esitatotus uumcrusib ctus pars et cpars b t numer^ ocnom inis bin afit oct oenominasc in b fjtc.fircpf numerus qut fit epouauoin e. ©ico crgococnomiari inaab fuumerofC5 qmfit ep ouctu ruoium oenominantiu vntus iii alterum. Dtuioo enim fin partes equalcs c:t ftmilircr ain partes cqualeo b.et qt quoties beft in ailoites ecft inf. cft enim occnoiatoi vtrmfcp comunis per bvpotbc fim cum bfecunofi tpfum numeref in 3:« efecunofi eunoem numeref in f.Joeo quotpartes immer 0 babcf 3ao bitot l?abet fao c.at qi cnumeratbfccuneuc cum ct bTpotbcfi efir Oenominas quoties c in b.numcrabifigttur cpiima partem a cqualc ipfi b per pilmapartc5 fequale ipfl e.xeaoe ratione fecunoa per fecSoam ttertia per tertiam.et itaoe a-terts fi plurcs runi.quarec per occiml numerabit afecuMu f.per Diffinitione igitur Denominabitur quoties cin afecunoa fatep ferit ipfum oeno» mlnans.eft ttacp cpars partis oenolatapars totius aab fnumero quioe qui fit cp ouctu ouon numc rotumiUas partes oenommantifi vntus tn alterum quoo intenoitur. CSi fuerit ptimus fccuDi tora pars quota tertius quartiutecp fccuDus qumii quota pars quartus fexn: ent ptimus quinti tota pars quota tertius fexn. eSintfcc numcriia piimusib fecunousic tertitisio quariusic quinmsif fcprue.fitcp atota pars b quora ccit 0-tb tota pare equota 0fuerit f. oico igitur atota partem efiee quotapars ceft iwms f. litcnigociiqniinas quoties 3inbetb quottesb in e.cum afit parsb Tb parse. oucafcBg in btM‘ ucniat li.qi igitur acft para bpamsfcj ipi*e per pzeceocntl aoenominatur pars eanumero Iz qui fir epouau ginbnumerozS illas partes oenominantiu vniusin alterum, at numerus bocnominls btn e eft etiam numerus oenominas 0in ftg oenominas ainb etiam benominatcin 0.Denubitur igiturpereanoempzeceoentemcinf abeooannumeroUquifitcpouctug inbocnominantiG illas atifi pjm8i)t33requoti pirseft apt(mu9 1quinti tota pard di cfcrtlus fforti rrpotcque ab eoccm nu* mero Uoenomiflatunquoo dl ptopofitum* CSi'p!irau9mfcamDoquoti'c0quartuiJ fnfwtq :item® tcrtfuem quarto quoncq feSo mquintoutccdTc dl tonco efie ptirau in quinto quoifco fuerit tertiuo in feno. CSti>t'ttp4n9fecRumeri;ap:finus:brec&&ue:ctertiu8:pquftrtu8:c quintiisifrcctus.ITtffiatora parsbquotaoeft fete tota pare oquot3bdlipflu9e.8icoatotam partem encequotaceflr.bec pe tMtue Tt pteooene Demonnrotur. CBi ftioi tp:imu3 iota pare fcdi quota tertius quam’ :itemep paimus quinti quota ternus rem ent pumus tota pars fecunoi et quinn quota tertius quarti zfcxti. fet numerua pntnue:b feeuouerc tertiueto quartaete quintuetffertua.litcp ptimue infedo vttertius iH quarto ct pitmue uiqumto vt temua tn ferto, ^tco aptimfi cife tota partem bcfccimoi «tquifi quotae ternas cft ofquarti » fctri.qi per l?rpori?eflm ptlmus ia feefioo vt tertiue in quarto fU ergod numerus ocnomtnaa quotice 3in bct emo. tq: etiam per canoetn l^ypotlpefimptimua lii quuirovt tcmu9mfecto.flt ergo bnumerus ocnominis vtriufi? fQ quoties atn etc in f.fitep ?po= Urum ergip numeros UUDncflcqolti er ouetuamgetprtantlt dl^tumbae.ecper nonam quoPitt cca[ng 3b:t3mfien^mqde):ainUeomporitGer^fle.Stmilitcrquoofitcreinsctl^: tantUcU q»tum 0 T f.et ftum qtf fit er cin Itagitur apitmus tota pars cft bcfccunoi et quinti quota eternus 0fquam etfernrvtqiiinumeratur luiUismeunoem numerfi U:quoo vultpzopofitio. CSifiimtpamuofccunoitota paro quouicrtiuo quarti pjimioqnioc minonbuo cmpnmus tertii loia paro auiparteo quota paro aut parteo fccunouo quarti. €Sitap:fintieniimcnje;btcfca]nwi8:etotcrtm«:cfquamie:3ibciImule):inctlbusmtnoubu8 ctquota parsacft be tota parsD ntcf.oicoigitur quarta pars autpartcsbcfiientcftotam parte autpartcsiramcrDacfrco.DiutDoentbcinpartcsfuasequalesaqucfintbctc.itcmoiuiooc ftn partc3fuaecquaIc8 0.ctq:cfcftequcmftipIctaoovtbca8aeEbT^oi}?eri:'it'co tot partes numero cruittipfmscfquotpoftiefuiitbc.Sintigltur lUcezf.eum igitur bpzima pars beftccqualis ate piima parstpfius effic cqualis 0quota pars aut partes berit etot» pars aut p.irtcs acft o.ct ita pe parte crcfpcctu pams fet reliquis partibus viuus ao rcliquas partes alterius ft pluree ciTciu. Jns tdUgcigiturbcircp:imttnumcru:cfceunpuin:ctcrtiuni:fquarttJ.ctfibeftpartC3C argumetozpcr quintJrqiptimiisbpzobatuocftrotcpartcscfccuoiquotcEtescftctcrnusfquarti.crgobe pzini'» «terruis tote partes fiint cffcoitipi tquarti quote part» bprimus cft cfceuPi .at aao p monftr.ar’ eftforcparrc0quotcbeftc.igirurctquofebcfccunou8cftcfquarti.etftbcirctparseomninoconfij miU mopo fwquartS vt nune quoi? factum eft perqiutam cUcerem argumcutu. ir>anifcftu|ita<p cft nprimos in feeoijpo vt tentus mquarto ptimisleriftentibus minoribus :prunum totam partem aut partes cITc leroj quotapars autpartes fccunous iquarti. C0Ipmmus tote parteo fecunoi (?uotc tertiuo ejuarti ptimo zfccunoo minozibuo erit pntnuo tow pars aut partes tcit^ quota ps aut partes fccunous quarti. eSintquMtuoruumetirab primnsrcfecunwjsrn eterriusretf quartus, quorum abtcbuo primi cuobus rdiqois itat minoiesret fit abtote parteo ciquote pecft f.pico ab totam parte vel partcg ee ocquotaparsvelpartesceftf.oUiiPoenlabtnraasparteoaocquefmtaTbetfiniilircroct fuas apfqiiefintocte.cunUgirurpcrl?Tpotl?efim3btotp3rte8fltcquotparfe0©ceftf:totcruntntc« numCToaboiulfiquottoe-ctapiimaparsabtotacritcfuitottusquotaopzimaparsDeeftrGiffl nusf.quarepcrpreceoenteap:im3 parsa baoo primaypartem bc tota pars autpartes quota aur quotecaof.ctfinuluerbfccuiwaparsprimiaocfecCoampdrrcmtcrtiftotaparsautpanUauofa vel qMtccaPf.fictppercUquisparribuspzimiaoreiiquastertii fipluresfintpartes.-Btfiaabofit P^6faaoaprimumn«mcr2:DfccBnmim:btertlum:tcquartum.etargumcntoipcrquartam:Quia aroraparseltoqttoabefte.ergoabaooctotapai8quota8aoo.re9aaoomonftratuseft{totaDB quotaca©f.ergoabp:unusaoocrerriumtotapar8quoracapffecunou8aoqu3rtum.(^oofia a» 0fit pmcBpenitus conflmili mooo per quitam argumoitabcre quomopo nunc per quartam ot parte conftat elTcargumentatu.T^Unum itaqj cft foUpumefie pzopoftwm. £tfi centtnctcrct puoI piimos numeros PttobosvltimiseircmaioresrconueifomoPocondupftmmclTctkcrrlumDrimrto- Mtn partem aut partes quota quartus fecunoi. quoP facile monftrari poteft omine permutato com lanjcnoofcjouoovIrimoBprtorcsetprimum.-rrccunpiipofteriorcs. ‘«wcon Crinionumerus mino: aut tfiaiotis pars eUiaut co ab maio:e quoties poteft octracto relinquetur iplius aut pars aut panes CSintabPuonttmertam3iorctbminoi.oicobcircp8rfemavcIeooctrartoquoticeporeritaba roBOJicrcipfins baut partem aut partes, fi enim bmino: numerat amaiorem per Piffinitionf beft parsa-Ofccusamem bnon mctitaur.fiibftraao igitur b quoties poterit ab aremanebit mino* b •JismnnidrilrcmancreinumcrarctafccunounumerumPenommantcquoticsfubftraberctur.auo tics etrimaUquis numeros ab aliquo fubftrabt poteft :toties in tpfo numerabilis cft per rcttapcri' rtoncmqulfiremsnctmtnoibperprimampropofirionemcftipfiusbautparsautpartcsrquoocft’ paopofitnm. n 50 .C.Si qUolc partco pzimus fccunoi rotc tcrtiuo quarti p:imio uioc fcmp itiinotibuei quoties ent pumuo iti fccuDo toties tertius in quarto.ct quota paro vel partcopmi fuperunt in fccuoouora paro aut partes tertii fueerutu mquarto. IESitap:tii)U8numeruo^fecu©u6bcUcrtiu8»;«quami8erct p:timactbcmio:e8:(itq}a^inu9 totcrtorcpartcebcfccfioiqiiotctquorcEresctmma cftefquarti.EMco quoties asUmbctotic» oelTein ef.ctquoraparo velpartes apnmifuEfuntmbcfemoo totam BtcvfBtesPtmilrcmancrc mEfqrco.fubftrabo ctit aptimfi afrfo beqtice potTumtfutp totus fubUractue btrctiouue c:ftt« 9 tottc8ig:qt[e8atb.q[ioct9citaporiroqtaroteBte8ebc4te9eef:crgo BTiccfimaoaauIatotaBB velBteg eprista ps vel Btes bcccf-t^urfus teclligo 8^mC numerQ.b rc<m.e tem6.gquartfi: tpriB* argumeto: bvtccfitnafcptimS.q: atota bs cll b4t« b cftgB brpotbefimtergo bvlccflmafcprtma qta B8velBtc8bcg:eot3B6ve[Btcsaco.ataoiaeru9eUtot8p8v>clBrE8b4»C8TelBte9bcetlef:igf quotaB8vclBtc5cbcaocf:totaB6vrBte8eb30g.atq2 4rBrc8albc:totBtesocef.etquotBte8 acbuotBtcs 9eg.n3btgcqmultipltcc8:etaepau(io:E8toteBtE8b^totiu8bc:igit-xopaucio»s tote Btc8 cg 4’ cf.fubrtra^at igif gab cftfltip octractuse treftbuus ftt pftobii pttmU^Jponis^tpo titu.fcj 4Uc6 ambetotiesoee inefcuefite^lis g.eibbt en3ntanifeUacUbtonBi^^Brtec0ee '1H3 etgequanf:9ctracti fcj ao pecraa& qta parsvet Btee etotus bcao ef totu.n aQt n&erus bipt^ ctotaBsfucringBbrccrima^ma erit creTtsuusireribuif tota B8 4^3 tot^totiU8.naitttoteBte8:ertt Eviccfim5fc6a5crcfi9u«»toteBtc6reriDuif:4tebctotU8cftoti-’.f5eUogvtbtotaB«rue4t»B8to^ fU8 bc:c tote’ cf.5ntcUigo cpmC.ffcOnub ctertifi.ct efquartfita argumetoz bviccfimafeptlma, 4a quoraB8tcf:totaB8bceefT>tnEoUratucR:ergo4taBBvelBtc8ceaPbf:totaB9velBteefeU30e f.etrtbcetDtceeofirrargumctarcBviccfimaoctauaipcluocitef BcactotaEtemvelEtese^totius bc:quota parsvcl panes fcfl totius cf.et l?oc cft totum piopofitum. CSitoticoocmptopjimoDcrccunooquoticstertioDcquanotOiaparsvclpartca ptimi relinquatur mfccunoo quow aut quote terty in quartoiptimi! totas ce parteo fecunoi quotas ternum quarti ncccflc cft, fril^cceUpuerfaptccePeris quevtpcetictESDuobusptimis n&eri8miRo:ibueintdIigaF.ririgitura vtptius pmus numerusib cfcefipusio tcrtiuslet efquart’’: et fltit cuo pUmt mo«8: ttoties oetnaf pzimueafecuoorttcpc^tus b:quotiestmi^aquarroqu(D^tusateetrim4t8P8r8Tel6reejpmi reitnquumrinrcOototaparsvdBtestertQ relinquanturinqnarrotfitiscreUaafievdBiesinuSa ecf in quarto.DicoergoapzimU totas ecparresbcredloiqnoteo tertias eaefqa8rri.£apiogqttt fittoteptesef quote Btes aeUbc3BPzeceoentemquotie6aeflinbctotte«gerltincf.etqtiotaB0 autBtesaruBruntinbctoraBbautBtesgfuBerfittef.Srtotiespoflrusdtbtf lnef:4tl^aeUin bc.cttotaEtemvelEtesbfuBctrcineftquoraESPelBresafuBfantinbcigtturgt oaoiotcemftmt cqualcs.quotc igitur partes aeftbctotcbeftcf.quooeftpzopofltum. CPtitni clementotum aritijmetices 5o20ani finis. iRopottio cft ou3¥'quantatum clufpcm gcnenovniuo ao alteram certa pin quatiiatc rclano.CTBumcri ao numeru oicitur ^pottioiminotts ejoc ao maiotem in eo ^paro cfi: vcl parteo' :rnaiono ^>0 ao mmotemfmqp cum cotmet ci*cius partem aut parteo. CjOtopottionalttao cft filirnoo ppouionu.CtCoiinnua ^jpottionalitao cft qft pncct0tur n6 oiflbeiarto tncouo.bcc aut ao minuo Iter treo coalcfcit tcrioorcu faltem vnu Tumcoum fit mcoiiJ 5nc6imu3:&o cft in qua mtcrccoit mcoiom interruptio. £t bcc ao mlmuift quamio: cpigit tcrioo ppier alteriuo mcop fmnpnonciii.C^uaoo fuerit treo numeri pmiuc Dpo2ttonalco:oicctur pnmi ao ternum p:opo:no p:imi ao fccuoum Duplicata et ao quartum inplicaia.cCum aut cotinuate fuerint vcl ccocm vcl oiuctfc,ppo2iionc0: oicctur pzimi ao vUimum ^jpottlo c]Coibuocoinpofiia.©enolano oictturppotnbis minotio quioc ao maiozem paro vcl parteo quota vcl quote tlli'’ fticnmimaioao ao mino2cm mimeruo fm quem eu cotinct ct paro vcl parteo mmoiio que in maiotc ftipcrfluunt. ©imilco fiuc vna ali| caocmioicutur ppotnonco que canoem rcapiuni ocnolanoncm.maio2)&oqucmaio2cm:ctmino2qucmino2cm. ^ ,eSi fuent ptopotrio ptimi ao febm que terti) ao quartum: att ccoucrfo qucfccuoi ao p2imum caocm quarn ao tettium. CriOccbenioflrat^ucrfam i)po2tlonalitatem:qucab etKlibcquftogeomctrieponff pzia fp^s^ipot rlonalitatum. ct cft quoties coduMnius pns ao anceoenert piis ao^^s: g» afis a® ans fefe babuent Ttanccbcnsao ?n0.aficc9C9 enl cftpUmfi jjpoitionis ejiircmumzpfisJi-ofctfm. Sintitatpquattuo; numen:apnmtt8/bfecuncu8;ctertiu9(0quarm8:ftt(5«quc|wopo2tioaaob:eaocmntcao©.eico ecSucrfoquep:opomobaoa:catic5eireDabc.Tfl9mclieafitj)po:tiocBbt>qucaaobEbmmitioRe eanoemljabebfltPcnomiationem.fiitBi^afitminotbeyPiffiiiitiolbustotaparsvclparKseriiaa» parfesJmanifcftfi cftcnl cum &ITt numeroe p^mue/ml parriG bcfrnitfltS.caplo fr0ovn3 partiC b que fltftctvnl parttu cq fit g:4aigiEquou pars fvnitascftowfa pare griKpfiusctergopcrfcs pnmJ ptiml quora pare fvnitae ao gttota pare oeft ao emomerabit igif oipfuc qtf trcrC c^poflrit rSj mimera «ouobusjJOuctu tertius numerrtuoe alteri illo? ent comcfurabifo CSitamimcrtiCCTbinc^roucrue quenuiner«c:®icooalrerfnumcreBbautcdre comenfurabi Im-namambcftprimuetaui coperitu8.ftp:imu8'.crgopcT pceoctiti altcrfieoanumcrat. quare al» rrri cos mtpm^urabilie.ri d^oitraetergo per fccnnbd buiue ab aliquo ptimo numeraE p: qui fit e.4acnatticrato:Humerabtcettapcmccftm5fcrrt3p2lmia:quarecpcrpcc&entcmjmcrabitbautc et cfictiico'ternumerctb:funtpcrolffltilfloiicoct alter ouopbautcqaoecoircrciocm numerus numcratKomcnfurabilecTquoPintcnPitur. rSuptfo quotiibct numerie pjimisioiucrfu ab ^U^a nomeru pjimu effe ncccffc cfi’. CSint ab cquoritbrr numeri ptfintiofeo alrerfi Bbipnsefle.Emcoenliiumcrflptimuinfccfitifi cr 4)nt3tUtb:ctrertifim^Piicrfict^!ciat cietflcfTerquarfuemucetelpffiin^ibutf&eictitabcinceps ctquiaptiotcerrmp erGiparrceparrifivIttmijioacriiergopervtccfiiitatcrriSprimiet otffwtiricnc: oC8iIUnummpziminumcrabiirvItfniG,poucrQ:cuiabOovn(tatcni:et(Tttotue4)Ouctuef:qui aufeft priniu8 Tictpc5(labitjjpofttfi.autc6ponruB:qu(flftJcrltcopofitu8!pcrfecunp5 buiue ah aliquo p:l mo numerat qui (Irg.ctoimg (It pumiiecf noncali4s pplnoEmamiialiqufepPictoBinumerarct ffoni nemeret ocfracffi e tt cflmfg eflqjer ouooecimS primi nomerarcf rcfibuu: numer’ fcj rnita* tan/quos dlimpoOibtletconilat ifftf alifi ab ipile ciTc numcrtptimfi. rSi fucrititouo numeri prrafcptimit^cucp numerat oltqicftao rcltquupnmus. <ESmtaet bpfrarcpiimi et cnumeret amifobtcclTcpfrafcprimosraUoquin IT enumerat bflut abus qiitcfii^b cr cteum icem pcrviecflma tertia ptimi numeraret arno cfTcnt act bcontra fc ptimt quo» cft contra bypotbe(Im. Ci^uiomep numerat numeros centra fe piimosict ipH funt aomuicem p:imi CSmtabnumerafeecpnnmcroPcSrrafeptimoaiaquibdcetbipfiSctPicoabcccotrarcptimos nam ITITnt comcfurabtlea fit cco« coiter numerasuja ergo enumerabit aetia perviceiTma tertii p:l nunumerabit c.ct4acenam numerat bpcrcabcnumcrabtto.nSffitigifcti contra fcpumltqficft contra brpotbellm. crflumcruscopofTtuscTCbuebusniimcnspm'! fcpnmisiflDvuiJcpiHe:? cllpmus C^itt at bnumeri corrarcp:imi:cTcopoii.1tc:t>i:ocaonTii(pfOBeircp?lmfi4n3mfifllccmenfus rabilid 'micos vt nuero a: numeret igif cos aUcjs mimcrne ft oitja ergo t> numerate ci aPeiractfi: ergopcrbuoPccimaptimiPctllnnmenifbrcncufi.norutigifabqtrafcprimi/qtfcitcorraporttu. eSiDuozum numcrojiimvtcrcp ao ternum fuent pnmusict qui cy ipfis pjcouaf abcunocmcntprimus. CSintabPuo numeri quopvrcrtp aoclit|5mu8:cffl'tbj>buctoseiraiiib:bicoOcirc^mfiaoc.n3lT cctoeiTetconiefurabilesialiijseosvtccotfcrnumerarcttqutciinumcraretP^PuctfiejPuob^fcja et brp quarra bui^colcabit alteri eo?: q4 pfeptima bui^ iipofnbile.n6 ira<9 ett bfQt pmefurabilce €Sifuerit DUO numeri ptra fc^mi^ cjralfo ifc Ducto^Ducif ao rcliquu erit pm** CSiiitaetbptrafepmiiet ain reoucrus^bucat cmicocefTe^mdaD b.capiobequaleaiet 4avrcr cpaetbeftpiimueepbicTgeperpccbcnteqmfitctainoeftpmusaob.feoquifiicjtafnrctequaE ct qui Gt ej; ain bugitur qui fit ef: ain Te e(T primus ao b.quoPdT propofitum. eSifuerint DUO numeri contra fcpnmuqui c?: cis mfc^JDuccnf crut aDiniiiccpri= fi iiDcm in^iDUctosDucanf:ct qui inoc ^iDuectur erunt cotra fe p:imi.5iccp ocmccpsfiprinapia mpDuaosDueannirt^Ductieriintincommenforabilco. fC&ntaetbcotrareptimimucafcpainfeetfiatcetblnrectfiatDmicocPcfrccorrareprimos. ^temoicoiiabucafincetfiatr.et bin Pt fiat ^rinrcrcfecincomerurabilesatt^ cotra fc primos, et ffiterfiouterisainectb in f;qui^PucenfelTccofrafep:imo9:ctbocquonee feceris ira cuenirc. ')^rim&patet:namperpmiflaquiaacflp2iniusapbercfite};ainre:per pcePentemceftpmus aPb ctquU ctilprimus apbctpfitepb in fecperrapcmcefl Ierimus aopquopeft ptimfi.'^:etcrea4a vrercpa'etceft primus aobifgitur per pecimam equi epipUsproPucirureff primus aPb.quiaigiE ebfunt contraicprimitet PproPueitur epb infetper precepentm eetia; efl primos aP P.cum iracp vrertp Puorumbofltprimus aPept''monnratumeft:_{giturpcrPccimamfqui fit er ipfis rdliecrer bin pcfi primas aP e.et itaqnficuncpfcries ;pcePat cocluPc.erbocefircPm. f££t buius prima par< re cognorcifrtetragonosquo;. latera incomenfiirabiliaCunteireiitccmenrurabiles.etcp fccutaucr feras incomcnfurabtlcsefrcquarfilatera funt incomcnfurabllia.cr cotra Pemoftrarcprcmpru cfi fi l9rerttfuntc5meiifurabilia:tciT30onosetfc(rcra8cnce6menforabilc6.9rfmt8blafera cemenfus rab(liaetctetragonu8a:eteeiusre(rera.Ptetragonusb:etfciu6rcncra:qtiiaabfutcomenrurlb( lesrcrgoenficruscditcreos numcrlsquifrtgiquiagnumerato^crgopcrvicefimatertiapriminu' meratceictetquia cttlgnumcratb: per eanPcvKcfimarerrtanumcrarPetf.fur igif retragonicp: iHPeetrefrereefapmuicecomenfiirabiles.feppcbisbacrcnusrquocinfottobuiusvbicenumeris tum qnapratis tumcubis agetur e^actifTima peterminatio futura cfi. ij csi0110 numeri ao aliquoe ouoe rterep ao vtrunep fuerit primuoir om eje ds p2d Duarur ao pzoouctum oc rcliuuio ait orimus , tI.Stncabp:imlaocctp«mtat>0:TerainbfiatcTcj:c(nofiatf.0icoeetfefrcinttrrfei)dmo8. ni «1 qi werqjnumerus atbdlprtmus a» eper oecinu butus c«ft pamus ao e. et qt uerii werqj atbelt --*i pttmu8aDO:ergopercanoeedlptimu6aoo.ellita(pvter(pouo2umcomoiiraruepamu8aoe:qQ8 '’o— rc per canoem oecima fpzoouctus e?: illis eft ptimiis ao c: quoo eracoemotlranoum. 14 eSifuerint ouo numen contra fc piimi minozep oc maio:c aliquoties octrabatur t aut relinquetur vnitae aut numerusao octractum piimus (E^ints bet ccorrafepriniitabmatot/cmtnot.oetrabafcpcocab aliquoties: Rnpfubflractas aet rcfiouflb.ofcobauteflevmtateautcffcnumeppiimttaoc.jBamribfltc5menforabtliscfttoUJotfi cotsmenfura.qtenumerat cBviccrunarcrriamptimicrtainnumerabita.ctqt etiam tromeratbper nona pilminnmcrabittotum8b.n6cruntiffitur ab tccontrafdpzimitqtfeft contra bfpotbeflm. G£umfuerint ouo numeri contra fc ptimi ct mtnote oe maioti quoao pot tKtractot refiouu apnus octracto ocrrabatur cotmua boc pacto Detractione facta :vnitate rc* (inqui nccc cll.q’ flvnitas rclloua fucritipolttos nOeros mcdmduratulcs ce pucnict. dSintaboiio numert'cotrare^mi:amalotetbmtot:oetrabaFapbab8quotiesioi!erip2tctrdia quate qui oetrabat quoties potanuero bft itacontinue boc paaooetractio quoao pot. oico primo tanecreliailtriTnitate.oicorct5oq’ribocpactoc6tmu3taoetraai5ercUct8ritvntta8 0ato8nneros cirecdrrarepzimo8.'iptimSp5.namoetrabatboeaquotic3 poretreUctfireTtoufi fltci^creUctS minus cb.tEiptcccecnteautmrnitaeautnficruspiimus 80b.rirntta8Coilarq64)ponlf.flnfiem* pmusaob oetraboipm quoties poiru5 ab ipfobrrclinqturmin’cqSrito.quoo4ogpperpceoentl ftireflmitas autniierusptim’ aoc.rinfierus^mus aocipmoetral^ab ctqoareautfemc oecreTcet numer’oanoommozcaopiimfitme»:igentia pceoet(e;autrel(nqturmita8qi5eftpilm£.Sc8mp5. nafl3etbfuntcdmerurabt]e8iDtnumet^nQer8btttotfitDetract6.qu8rep9uo9cciniiptimiioena mcr3bicreriouQnumerusrc5mit8t£qi5eflipofTibile.qtfquioevt clarius Bcipiasricpfloera4ri8etb nntpmtfurabilcs D:cr0o eos cotternumercttrubflrabatcpb ab aquotiespSt et reliquate defu vni tas cfi numei^ onumeretbnfierabit oem numeratOab b fi imi: quare totfi oetractfi ab atnfierat etia avtponcite(l.er0ocouooecim3p2tmlocti3numerabttrtfioufifc;c numerus icjmitattqSefl Ipombile.rflcittnfier^qzoiiuerattorSaetOetractfiTt^batfiexrgoBOaoOedini pmioetllnmne rabit crcnoue.Detrab9f ergo quoties p3tcocb*relinquat e. il edi vnitas argurattsre Pt pUos no merC 0nutiicrarc ctmitatt-ri vero efi tnficrus conflOo iprum numerabit, et (uftxoioo fl» qnc6qsoabiturvmt38 ip3monumerabtt:q0dl&>ofnbtle.£teicbscocm3th’«ioi)edittftouoppini rurabiliOaltcrabaUeroquonespotfcibllrabafnKpreftPnuorefiettfiUlisefceotDesfnraljlIt. Itf C©ansouobus niuncris'c6icantibu6:madmu cos numerantiif!iicmre.wwindm fclM cft q» ois numerus ouos quofcunip numeras majcimfi cos mimmn^ nderabit CSintabOiio numeri coic8ntes:3n»ato2 etbminot;ppofitfienma)cimBnnme?ipfo8jwmer3tem Iuenirc.Sicnibnficrata:cabfecti3numeretipfcertm3rtm<»co8numcr58.nn8UKbtwnumerat a oetrobatbabaquotiesptetfit relictus c4fl nueret rtrScy ipfe enumer*pettomJt no n&eretoeira batergo itcrilcabbquoriesptctfitrcUctusoqeritminozc.finumeretoJvti^btaipfeepetlms* at fi no numeratqz no occrefcct numerusintnftnitB nee$ occumbet aovnit^qzper pceocntl act b eirent cotra fc pzimi tanottenenief ao aliqucquian rclictfi numeret qttc§>pzlm6 ba^iroeoico eiTcmajrimli numepeoscoiter numerante. naficoniaiozponaftacdoetDOUooecima pzimi inaioz2 nunierarcmtnozezqScflipoilibilc. £j:botcozrctarffieflnotum.pon3tenlabBumericolc8n*e8etb minozfubftractJ» ab aquoties pot treltcms ext cfubih-actus ab b quoties pot t Mictus o4f^Unz ctusabenlAilrcUnquatrqioeoeritmatimusnumerSsatbper pftt<m«tntff^Uwnum<r5s8b. eicognficrareom3j:im6numerantt.fl cnigfitiocnumero oicumoienmnerus Copmnumerete^ occo vemmertpzopofitii.fi eft alius ab o no erit malozo.nSonoeiretmaputuisiuuncrtts numeras abverfi potiusg.crir ergo minozoe neccfntatc:qui ({anumerat b. ergo fi vicctuuterRam pzimi nu merabittotuni ocrractd ab 3.1: cum eti3 numeretazper euooecima pzimi numerabit reuouo ett cum numeretcpcrvic^imatcrtlam pzimi niimerabiz totumoetraciBa numero b.ctcnto uumerctbzergo perooooecim3pziniinumcrabit reriou6fc50«iuooc(lpiopofitBni. .* ‘7 Ctf»20pofitis quorUb5nueris aoiuicc ppofins mapmu cos coiter n0crQte rcperirc. srsintabcquotUbctnficriaoinuictppofitia)pofitfienmatimanumcpeosn6eranttinuenire.5n oenioentper^etentltnattmannmei*e8ctb4flto.4rimijneratcdUeequlpettfaUoqulfloabi8 maloicacciectfiOuooedm5pzinjlmtnoztnumer3remaioz5.fi8fitononuiner8tc:qzo0oqcuncpnB mcratabcnBCTatab.tiicunipnfieratabficozrdarifipzeceoftisnnmcrattmaiamfinumepeosna meranic.er«o 01cfunt pmirur3biles.lnuenio ergo cp«ceoentemajiinfi numcp numerantec104 fit equeoleo cITe matimfi numerantea b cuta ft fit quicfiqj aluis malozecos ceiter numeras: is fit f. daergo fnumerat abfi cozreUrifi pieccoetls numerat 0majarafi eos numeraiite.t4a numerato et eti3cpcrpofitfi:ergoperioccozrelartanumeratemaidm&eoBnfierante.atf ponit maiozc.ntterat tgifmaiozmlnoze:q6cftlpombilc.reltn4mrigiturcdreni3rinifinBnHT*nteabcctpzoponttt. B- b- I 8——b -- c C^uilibet DUO numeri in fua p:opo:tionc mmimitfuntcontra fc p2imi. C^intab ouon&eTiinruaa)pottionem{mi:oko eos efle contra rcpmoe.nariflnTppoTiri fit cois eop nicfurs c4nfierct a^tn oet bccreptim^fct^i qjipoulo oao eca eft aao b.at dlb Tu mino: a et eminotb non erant atb in fua a>pouione minimi:quoo efi contra pontum. CHumcri quiliTict in fua pjopojtionc minimi numerant quoflibct in cupcm pjopoj „ none fumploe minot minotc tmaioi maiorem cqualitcr. CSintaetbmimiinfuajipoitioneamaiojbmlnonflnnpciufocptopouiomecOfcquioSmaiotb m(not.otcDanumerarecetbnumerareoequalif.naq:Ttaaobltacao9: ergo permutati per tertia fecfioivt aabcitaoab b.quota ergo pars vel eteaefinadc tota pars vel ptcsenbaDo.fttotapars confVat^ipontfiaequalrnfierarecTbnamerareo.fltotepartescrgoEtricefimapUmiqttotiesertt alncqnorte8bino.etqnotaparsvclBKSareniafcrintlnctot9 parsvclBtesbremancbfitino.fub flral^o ergo quoties polTum sbec toctract<Tlt eet refibuus f.ftmilif fubftrabo quoao pofTum bce b etbctractusntgir^buus b.qtergoanGrratc:tbnuincratgequaIif aticftn cunbenumerfi.ergo per octau.^rcdi quepwpottioaabbca efleabgtergo^iqptopotnocabb totius aototGteaefleabg bctracti ab betr3ct1i.igitur pquin^ fecunci q;ppottio cab btotius ao couin: ea cft fac breficui ac reribu6.qu3reiterumqptopo}tioaabb:earnfacb*sccDmfctb<’^nctarintmmo»atb:noerant itacp aet bin fua pwpoitioneminimitquob eft contra ergo quofcuiB in fua pto« ponioneminimos ceteros vt oletumell numerare. esifuerintDuo numen ^itiuice ptimi ipfi erunt in fua ppo:nonc minimi* lo C^intsetbnfieriaoinuic^ptimitamaiotbminot. bicoeosinfudippottioncelTcmimos. nSfiTint ali}4nntinni3j>po:tionemiini:ffntcrgotlUco:q:crgocbruntinUlag)pottionemImi ppccccntl nomerabfit quofl&et fneacS^ottionemtnot minotetmaiotmafoteequalif. nfieretcrgocnumcp afc^metonumeretbfecuntmcuneent numer&emanifelMcllq> enumerabit aetb: quare nofunt atbcontra fe ptimitquoc e(t contra bypon^efim. Cl^oUtio ouobuenumeris contra fc ptimio li alter Ducatur in numc? que reliquus no numerat pjoouctus quot^ ab coDcm non numerabitur. CSintabnumcricontrafeptfmictaoucarur in ctptouCTiatoetb no numerat ctcicobnonumcs rare e.Sicrgo bnumeret emumeretergo fpfutn fm efntcUigo pzimQ numeru a:fccuno& b: tertifi e etquarrfic.q:crgoqb cotfnemrrubcftrcmisaccquficflei quoc cotincturfubmcDiisbc.crgo per viceflmSquirtta fecunci quepzopoitio aao b:ca cft cao c.q; ergo abfunt piiml per ptcceccceni funt (nfua ptopotrione minimitergo per ante pzeceecnt^a numerabit e: tbnumerabit ccqualirer: quoc eflcontra brpbtbelTm.poflr(: entm erat bno numerare c. crSi numeram usuobuopjoouctum tertius numeret: loera alterum eotum numc< u rabit aui eritrtrieB commenfurabilis. CSitanumeruspwbuctuseiebincquebnumeretTbicoonumerarebautcautcfrcvcriiBcomenfu rabilem.Sieni alteri fuerit incomenfurabitisvtverbicaufabicii ergo betcilnt contra fepumizb incp:ooucata:ergoperpzecebentfnbnonumeratcneq$quibebnumerabita.atpolltueen:bnume rarea.numeratigitnroaUerSeop. SiitacpvnifuerittncomenfurabiUsnecclTarionumcrataltcrQ ibco vtriep incomtfurabUisHneqnit 4l^t vt aut nlieret alterG aut vtriep fit pniefurabilis t^pofitu. Cl^umcros mmimostPm p:opo:ttonc numero? alTignatotu pcrfCTutari.parciqp eje tj boc marimu duos numeroscoiter numerante: fmmtmos ilUus^^poziionis imcrarc Cnsintab in aliqua ^o:tione:p:opolltfiefl mtmos numero8.ppo:ttonis aao breperire.aut enim aebruntcontrafemimicttuRcpervicefTmlDufusiUifuntinruaj^pottioncminfminei^aliiqrenoi runt.8utr(mtcopofltittunc per becimafeictamDui^queromairinill numeru eoenumerantequi ficc nSeretqjafmotetbfnt e.Bfi^tfm3ergofc61eabecfl^poztioaabbzoaoe.bicoit3cpbetccfrenit mosnSerostn illa .ppottionc.lTentflntalii flntfzgiqzfetgfuntmtmiinilla^^Jottione.ergoBOsci minonSbuiusnumella b:maio:maiozS'Zmmo:mino:efmeunbenutiierGquifttb.iHicIligo ergo quattuotnfterosic tnimfiiffecunoG:!? tcrtiu5: et bquartii.qz acotinctur fiib ettremfs cb:z iod afiib mebU8fbtsrgocviceftmfquintafecunoiqp:opo2ttocabfeacfll7abb.ergoBmutati per tertia febi eaolefljipomoeaof^etfabo.feofpoftt^mmtotbiergo ctcmmotD-tficflc ceflma^im^numerue numerasabcst bpafttus efl etia numerare ab: ergo I? pcotrelarfScecfmcfeEre l^uius numerabitcmaiotf^ minota quoc ett tpofnbite.no cabntur igitur alif ab ipfls oc in illa ^poitione minimi quoo efl ptopofltSzet in botetiam cotrelarfS notum efr. C^uilibet DUO numeri mimmos numeros fuc pzopoitionis maioz minoze et minoz 14 msioze multiplicantes: minimu ab ipiTs numeratum pzoDucunt. £x quo condat mu nimum quS duo numerann quflibct ab eis numeratu numerare. Q[StntbuonfierfabmultfpUcatescomimosrue^pou<oni6maio:mtno:l zminatmafotSiblco ipfos minimfiab fpfts numemtQ ;pDucere.na qt fleut aao bita cac c; ergo per viceflmaquintafcSi qulcotfnetur fubeiRremis ab equuseflci^contineffubmeoiisbcquifitequembfcodfcmtnimG ab ipis numeratQ.Sl eni fit alter cflo ergo ipfe fqnea fmg^bucatet bfeSm b.poHtis ergo ozcinc fmfni8abbS:<lt<|uicotineturfubeictremi8agequateiqutcotineturrubmebii8bb<ergopervice nm3quint3 recuttotqptoponlo «39 bea cfll;a^^: quarect^caOD.ctq:ajT^ucUe^c5m9et!^ett^ 84>9ucitfrctfms>craocfepnni3fedioiqpzoporno 9aoff:eacf>ca9f.atc&cao 9riaiti7a9g:erg« emutarim ricutcaDp:UaoaO0^uarertnitcaob:>taeaof.reocnumcr3tbperoectin3Roiut7uius cr0o et cnumeratf:et fpofitue tSt minoremumerae igtiur maiormtnorliquoo cft impoiTibllet pro< porimm.£tet moooocnionfh'attoni8patetcorrelartumi 15 caiaopollns quoilibct numeriemmimu ab cis numeratu inuclltgarc. eSint abco quotlibet noineriiquo? piopontf mtnimfi ab cis numcrat&inucmrc.qaero primo mi nimBqueninumeritaerb:q> rianumcratb cm&eft beffemiRimS ab eis numeratfi.etfLactbrunt cotrarcpiimimico vnii in altcr(i.ct qr illi funtpmiicrsoevicerunabtii^erSc in fua^rpoitione mimi eprcceoenteisif Ructus ab ipfte cricmImusquenumerabfit.atn{pfifuerintco(cancesrumoper pceoenteminuiiG que numerat ci fu eictconnit capiSminimSqa^numerite e4fiifctminimEquS immer3rofqfit0queoicocgmiiiimEnumct:quenumeratabcP.numerabunteiiIeumc i5p:inu. at q* no numera minore p;.n3 firitoabilisnGcrosminotgque numeritabcbfltiUebrqt ergo ab numerStb.ergoe ccnrelarid pccoetioemimus ab eis numerat^nfieratb* et qr et cnumeritbrcrgo eio^ cozrelanuf mimus ab eis n&eratu6nGeratb<aqrfnEeratbetDpofit^efleti3ipnitnnSerare crgopcrtDcm cozrclarl&gmintmus ab ft enumeratus nomerabit l?>atbpofint8tn minor gmumc rat igitur maior minorciquoo cfl impoiTibile. itf cSimmimu numqt aquotlibctnumcramm aliquis p:lmttsflumerct:cuitoe quofp cUiqucmiUozum numerare ncceflceft. qC5nMteb’numerisrtf8cilcEpeiwif.SmtergoabeuonCcrlKmDnu8abei8numcrat’tett)nfie ru6pmusnumcraii6c:9icoDnSerareavelb.numercceniamininiEnumerat&crc8fne:ergoperz4- bui^ecrivnusniimoi^^pomdisaanbnumcrasb.tiicficonumerv pUm^nEerate^buedabuob^ nrtensactc.crgoequ3rrabniusonuerataItcrfinumeropaaute.riacoEat4)poatfi:rieUtoc.nEe v(cefimaquarunumerabitb>?n pluribus afitconfliy oE^itur.Stntabequoiltbetnfieriomlmus ab cis nEcratusicnEeruspmus numeras bicicoenfierareabaut c.capioenlfm etigentiSpeeo^is mrmfinumctrqucnumcratabqfitftctmimEnumcpqucmiffieratfcqucejcpccPitimanifcflficftcg 0et nEerct cmimn ofciStit g({ per 14tpuius eil vnus nEeropjpportionls fao cet 4numeratf. Cum itacpcnumeruspmusnumeretP^ouaEetibuob^^niimeHscgitrgoEquartabniusenEeratalterE numeropcg.ftnumeratccoilat4^oritE:rinumeratgergoevicefimaquartinumeratf.qiergofeft mimus numer^que numerlr ab.niicret ergo ammimEffcEm 1? 4repefompta vlcefirnaquartam i?m<'erttvnu8mimo9-^porRoni8aaobetnumer3sb<cum ergoe^zbarusiirnfierarefproouaEee ouobusnumerisal^rergoper quartam buiusnumcrataautb»nnumer8taconUarproporitum.n^ uiog.naml? numerat bi.quareepervtcefimiterttampnmietiinanerabisbrapropon^ CtCumfucrintquotIibctnumcri: 3Daliquemp:lml: etmmimttsqucmnumeranUD cunDcmcntpnmus. CSintnbcfluoilibctnumcripitmiaomfitqscmDnBsnnmer^quinninerSttbe^icoceireprima aob.nafioefi ecmefurabiUe ao c: ergo eftali4n numerieos cElrer numeris 4fltf-SlenIfnrnficr* pmueeunmuerctciergoBpccocntenficrabitabautc^cficrianficrct bnofuntergoabepimaeo qS cflcontra J?rpotbcftm.at fif fit nficrus ppofitua ergoe fcEam botus fab aliquo pmo numeratur 4Et g:qut Evicefimarertia pzimi numerabit0et e:qi ergo gcil pilmusa nfierat e. ei^o Bpicceoemi nEerabttabautcct etianunteraternofunt igitur ab eteprimi ao oiquoodlcontra bfpotbcliin. zs fumatur minimus numerus que numerant aliquot pumi numen aingnaii:aliu ab libs numerum ptimu: cunocm numerare impoflibilc cft. CGltemimus nucrus quenficrat abcquotUbet nucri ^miioico nullfi Iplls ^mfi alifiab ipfls nae* rare e.n3 ft carefaliuort e:qi Ipfc nEerarct e.crgo enflerarct erti abant c«id cU5l?fP«tb«Uni. i.9 CIRiimcmminimuiptopofitaruocnoiarionumpartcsbabentcinuCTiirc. qnSipiopofltarupartiumvtTerbicavniusfecficetcrtlequarre tfcjcteppofitfifitmlnimfinumep Etes Illas bfitem rcperire.capio abconfieros partes illas ocnoiantes quoymimus ab eis nSerat’ fit einfierctcp amintmue fcEm fiet bcanocfcEm gict ceunei fcEm 0cunoe efrfm Ir. altematim ergo Boctaua pilml fg1? Ir numerabut efm abco oatoe nficrosierfit ergo in epanes ab a bcoqui-zoico eiTcmlnlmuinquo Etes ille rcpcri3tur.q»enipaitc8 illas bnbeattlamnotfielhfeofl no fit mimus fit ergo illelmtnor eo lilasljabesit duspartee abiUisoenomlnatcnntmnop»cTOnt ergo altcmatimpcr oct3uamabctiparte8lbenomlnateabmnop.n5 erat ergo eminimas quenu* merabantabco.nam 1ponitur minonquoo eilconrra bfpotbefim. 50 eSumprts Duob'^ numeris contra fc primis quom mmozcoc maion quoao potcll Derraao refiDua Iit vmtasimultipUcem aiiufUbct eotum qui fuper alienus muUipU* cemifolam apponaivnitarcm reperire. CSintabnEcrl prra fc #mi:amaior bmtor metractoqj bPeaquoties potrellqiwf vnltasi^ofitG cftpareinrtipliceaoa:folaTnitattmultipliceaPbfuBantl.petnibo«’S®5tfipofrumbEtiatePctOf talia et ftt totus oetract^ aEri9ll6:4 qulo^elllpcautmftipler ao b.etfttcreUctamitasrappoaPb vnltateqfitP.ctpucoatotSmbpr^menlateqaemanlfeftficfteemttltlpUcStotaUsa.SictPcPuco p 4cbV f6-76 t<571 C4 e1<9 b!5 t> 1(58 a8 IIt i8 17 + 1(5 14 ^54 ^ 8^ 11 i8 4<5 3panialecunolfc?autitmlrtplUeaoblnboci 4)u«jUtflz<ru£mamfcfHicl>BtJprtmieemaiplicc flPbcrUU.lDari9ucocvrutat^inbPezvenutl:gi{ctzv»irad iiiquccurubnumraoucartlKt^Ducit cfleqiis btcfcs bofucat bfoU viutate>crgo Ugfuprrar bfols vniracc. H^anc i0ic Pemo qfit bstfic assr^t» fbril mftiplex a» btfe® aggregat’ fbfupcrai toifi fgfola vnitatc bg. fco c6fgfiat cp pu miapartlaUsTcbtbPcquafe4firetPU(tuatotaii8mboper occunSprimi.atPktfi effeee nuiltt pUre^aPatfb muldjplkeeo b. igtfemultiplciiapaequaliequioefgrupatfbtnulctpitce apbfola vnitate qti erat oempftranofi et propofttft quo aovni parrt.Sct» rurfum©emofh-anPu eft pabile effe moInpbcca&bqfoObjvnuAaupermuIttpUc^apazaaferoonifarSabbtotaUqfitoctPucobpar tlale tn tocaU aet prot^Utcmanlfcftfi efteei multiplice aP a.et duco iteru bpartiale in aparttaU equall ant mulrtoltce totius bct prouctUat fgmanifeufi cft fgclTc multiplice ao totu b.ocinoe ouco bDtn cvnttar€ ?rUt g b qui qi quicfii? numer^ in vnitat^ oucaFfeipTumjiDucit gbtll equalis toti b quaretorna fbeftmmpl^aDb.ScDcfi per noni piUniqd fit cpbBrtaUmacequcf ei(i fit epcoDcb pamaUinaetm ct illcqfireebprtall ma:etb totali mcfolivnitatefuperaDDatiLU^fitepbcttaU tnaetinc.fe(Jrurcrgoq’mnlriplc]C8Dbfc5fb4fitei:bpartlaUin8ctbBtiaIilnc:fol3aoD«vnital4 fuper emultiplici aD aqmfit e; bpartiali macquoD eflfecunDfi et totum ptopofirS. CfiJopofiae Duobus numeria coira fe piimistmultiplicg cuiufUbci co^i (nucfti0arc CIUU0nQ mulnpUcI altcnus oiiftrcna fit oato numero cqualio . ®^^?**^®“®'‘®*''^ff?m'*f*P*”iq‘io¥culufltbetiubemuraingnarcmulnpliciculusDuaaomul ti^«9l^»fitoatonaero4fitccqttalis.apioftnewgenti3pjeceoetlsmultipUciaitfitDctmulrt pluebqiiteaDDesfupermnltlpUciarolivnttatSt multiplico 9per cnuniep oatfictptoucniatft’c BcmDecet4menlatg.f qttifitftclne^mcinDetvnuatSpernonapzuuitolstiiierusirtvnttatc feipm ptoouclt«ergo Pt}ag8Dfeftnfierus Datus c«ctcS aflt pars Det DBS fit filFb pars ci: eparsg ergoBvuefim&ertiip^iaefiparsfetbBD g*quarcfctg multiplices funtao ab iniquos nucrus amgnat*eRDiIa.etp^i»nlFB^cDent3capere multiplice a(tfuBaDDcret vnitatfmRiplidipiusb. peinoej:cco«rewDtcroe».Cet^flminoi quoties pStocmaioieoctract’ nofoli rdlnquacvnitafc fCTdteriot fiat fi^toactio quo min’per Acuente opcTaripofTesbocmSpoicris operari-^atis eni ab noms contra fe ptnis et cnumero cui fit capifDa Diia multiplicifi equalis ct Detracto bab aquo» tiespotronaneat Det9Detracto abbquoties ptrcliqu’ fit cte Detracto ab dquotiespot maneat f etqitaiiDepvDedmiquuitabuius rellti4turvnita8fitergo vt Detracto fab equoties potcritrelin* Wtas.llcrgo muMplep 3DCbct aDoercfupia multiplice bfumo altcru ertremop qui cll in litu ipartabaTtfetfumafbmulriplccciustiaDpanmufupcrnmltipliccctiquofttffmgoucaFqsg in DetnatbUqaDDctrupcr multiplice cvnitatciqifiginoq^tum iiicvel ciusmuUipUcctuif. oempto ergo «10 De bbremaneat mIi vt lit mct vnitas equalc bflriu cin mUfm 14oucrus in bfactat ntcui a8oltoimofiatoqu€numcrabiiDeaDer6nctfitvtfc«mp4ouctusinafaciatr:qutmfaDDitfupcr mlfiplicebqui fitqynitatc.multiplico lgifrctqpcrcn^mepoarfietp;oDucatur8tt(^ erunt tnuU tiplicesb tatne feDita; rm enumep alTignatii feruatcs.jCcrtii elcmftop antbmeticesjoioani finis 5fticnntquoilibci numeri contiimc pjopozuonalcoiouoci ouo pwcimi cruni commcnfurobilcs . jCSbU^bcDqiiotUbetnfierlcontinnepwpoitloalesioicoabenecomcnfurabiles.jftJ ?bc etco« Sioii anueret bmanifeUil ell aet befle comcnfurabiles^et qi qpiopoitio aao bea ell baD c7cao 0 IwbD.quarecSIlabit piopofltfiiabib cc DielTe comcnfurabiles.ot fi anon naera^iqtipoiHbMefitbactbefinfuapiopoitionemimos.nafiaetb cficntmfuapponioiicmi ninUefient BMctmao^uiiter^aDlnuiefpUmhetper oecimfeona; numerarer quoflibet in fuapto poinonett^q P^<’^8ABb:eaenbaDc.aergonfieraretb^muscdtrarepiimuqbcfitpoinbiic etetaoppqflrt pofiti^eimtergo aet bIn fua ppouionemiml.C3pla ergo evicefimatcrtia jtertfi efmunosmilla pp^oiKqpct pedrninonfi eiufoe nuinerab&t abfmeunoe numerll cqualr vt fm gietconiurbeetcDannttgtf ab:b eiicDaDinuicgeontcnfurabilesiquoDefipioporit&i eSiimcro? coiinuc ^)pojtionaliu duo extremi fuerint coicanrco erit nucruo coiter oc^um^omattmurip 4oes numerabit ent mapmuo cCTrcmos numerao. Cj: boc coltat 9fUIiquot^>po:rton3le0 uif ouoe coicantcs ccciociititotioc inter vtrucp eo? ct mapimi numera co9 coiter numerante caocre necclTc eft.-paiac? ftt ctia mammos qui bmo0zbmos numerosi ptopwnonalm coiter numerat eltc connuc ^pottioalco esmtab co nficri ptinueppoitoales quo? a d eptremi fint coicatcsmlco pmo alique efie mimerfi eos oes coitCTnfierantf.fcbomailmfinfiep oes numerarecJremacimfi numerante ectrcmos.Cucni ppccoenteabcoicenttfifrbeetcDicapioBitftcTtilmayimfinuinepnumeratcabdfttctt mariinfi nflcramebc(?ritf.etm3>rlmGnfier3recO(illtg.pono(Bbbmunosini)poitioe3 3Dbetpert? tertii etipeiurDccfgnumcrabfitabcfmb:etgnqer8bitD^tU.fumoitcmacjmfinunicranteefdfUl7 maj^numerMtefgqfitmqierviccaiiatertiamtertfilnficrabtfctmffmb.etiterfilnfierabltf hnh 7miiumcragfcfimk.pcr.i9.tcrt«etUmfuntomcrfinfieriabbfi.nanioce(rembmrepioDu-' *«P*“Pn'^"«f8'ibtncpwbudtaetUingpioDucitbtbbpcr.18. rcrtfltunt contra 'c ?wIIt -®fuopcdfnfrerrii cetra fcpzlmiqd eft contra bTPDtbcrim.funt ergo 11malit abbMtergobUmimcratlctmcqualifinumeretcrgo ipfosfcbjnciper.^.tertiicftmaclniuseos iiuincr5equitie5nper.ij.prfminumCTab«abcB:qecftpamfitnp:op5netetmtff.£fp€rt)erinj3 fcpnm§ tertii ncritlnuctuflmajCinrtiomcrnenumeras ab co alio4nc cotrdartfi acdDetmaioii num«nmticrdn?mUi£):£.tq:{?pcBoiatnml*ime:crgo crjpiuTnnfierue4fit«blofeqwftt<* ocnoiatnfiic.tqzl?etMt>cnowte&i atcrgopeaiiBSij nomcm»qulfitcfl?to64iUQOcnomfiut tiina.rtcaocroHcqtUoenowtntnmctming.crgopCTViccfiinitcmlpjfmliturtier’ qirtfitethii» fcquifitpoenelatiitns:etertSqzl:PctiominatgtnB:igifpcreanoenumcni8 4firc^UlnpqiUfUr oenowtnutO4numer0ti0ituntnnmcrflafctfmqetnumcret>frttnr.«qibUfuntaOinaiccpjunU ergo per PuooccfmltertttqctrfuntcotrafcpumizquarcpcrvicefimaeUjfBeitifua^zpoztiotieintmi er con-dariiiigirurTtceftmetmteeUiroeqetrnunTcrabfitatorcbmtnaninfinmne? numeranti actP.c(Ksiturnmariiiiu8n0cnisnumcro8aBcjtremoenumer5eqttooeftfccunDfi.Tp>tlmfico2rcs larifll?incco0noraF.namqtio€numeru9binnle:p2ooucitlea.ergopcrrcptimapnmtaelnerunt cotinoe^ipotrionalee.tpcrioeqzltnamgpzooucitmgBzergoBgmnerfitcoRetnuejtpomonalcs quareficut inter aio caofitbcouomeciiaiponionalcafilVinterat ncaoutonoel.et inter Biixcas t)untgmouofneou4)poztionalc6rc3totiDcmeoit j)pomonaieeinfODOS effiontta et tottoem inter vtrfiqseopetmarim&coBcoiter nnincrantl:q6eftptimitcozrelarifi.fcgmverol7itic¬S.namqt per vicefimatcrtiam tertii ioecimanonam bmeinfet in gjjoudt ab c. ergo per feptimam feamoi abctefgruntcdrinue^poztionaleazqbeitfccunoQcotrelariumtptopori^. 5CTBumcroe oiucrrarutn p:opoztionii fumere :quoFum mapmicomuniter Wnoa ct binos numerantes fint cononuc ptoponionalcs. CSiiirabnSericontrareptimizamtnotctbmaiotiaiRfcjioucatcet inb 4>bucatbibinreptos cucatetper fepttma fccfioi qpzopottio a ao bea cflcao b.ct poetaua ctufog iterfi que^poitie a90 b eaefloaoc.funtergocoecotinuep:opoJtioalce.OucoSaincetfi8tfiinbTfiat0icrit«Breptiina TeeSoigabf ricutoao c.qu3reii‘icutbabaetqtqpzopo»ioaaobcaeilcaoo:ergoperz.5fecanoi qblTrepainOequiieftetqb ftt ci: bin c.atqb fit cp ain DporttucneiTeg^quooergoiite^binceflg* ct qtf fit eje cin 3eft f.crgo ecoirclartfi vicefimetertie tcrtiic eft ma>:im«’nnmeru8numeras fg.5teni OBcsoinci4»ucniatb5qtergooina^eucitg:‘roine#ouatI?.crgoBrcptimlfcbiq4)pomi)caoa eacft i?abg:etpiopo«iocaoanoeftvtb8ba.n3cibiorufc5Ctpar6cquarenF:quooeftrpolTibile. crgo^pottioipao^noen^otriogabf.JIteqtabninteorrarepmietepzoouciceebinie. igifcB vnoeclinatertiienpmu6aoa.funtigifetatnru3J>potrionemImi.atonfierarg#matio3on(ierat bftne.ii^FBcoircladfiviccilmefticFrfieSmairfm^nScrusnumcrSsgctly.fumocrgoUpiinifiaoo et malote 0in aUa quio3 jjpotrione $b ao act ^eao a:T oueo Ipm itmeet ^nicnlat I. qt ergo ein o facit et ein Ufodt Iper feptimS fedioi Q^ottio it ao peadtIao fj.atppoztioh ao 0pofita eQ alia a^pottlocbaoaeteao a.ergo 3aUa$gaDfni7aog.etqtboruntprrare^miTioinlWpotttone mtmit numerat blfmc.efttglFBioecotrelariil tjtmUemaidm^cruBrtnmei^betLii^tifunt igrfgblnumerioluerfapjipottionltqappm8ii:im(coebtnosibinoscdiarnumeratc8riintptinue ptopotttonaleszquoo eit ptopoficum. 4.CSinumero? continue 4}po:nondlium ouo c?tremt funt contiufe p^mtterant oH tUi mfua ptopotttonc mmimi CSintabccdrinueptopottionalesquoipaicIlntcotrareptimitoicotresniierosabceiremlmos in ;ppo:tioncaaob.nanitnt tres miotesfmtoefzqiBincefitnapnma Tegi eiequaptopottlonaUtatF qpioponioaaocea efioaofet aiccl7TP<3U7cf1mruntcontrarep:imi.ergoBnc^m3terttj funt in fuajppottione minimimumerat ergo numeri aeper oecimanonam eiufof numeros 0 1 fnnoiotesfc; itiinotes quoo eit ipoITibilc. 5CSiquoiUbct iiflcri ftiCTit mniaj)po:ttoc mimbipro? dup cftrcmi crui ptra fe C<!?ec^cduerfapceoentis.Sintergoabcinru3£pomoneminimi:oicoaetceirecontrare#mos. wlIaetcflntcoicantescritnuerusaU^sioemBiecunoSbui^eosocscoiternumeras^fuoquos qtabcrunitrcsnucrtinequalcsmOerabttfmtrcsoiucrfosnfieros.numerctcrgoafmeibfmfet cfnigzperreptlmSrecunoie fgfuntineaocm ptopoztionecumabcetdtnntminotesabcTtpoteti funt eotum partes no funt igitur abcoatiminimtiqeoo eft contra brpotbeilm. CTBumcros quotlibet Pm' Data ptopoitionc minitnos tucftigarc.Bfi mamfcftu ait cp DUO? in aliqua ^po:tione minimo?pumo in ambos ct fcdo in fcures. cc cp cooc pmo in lUos tres coDecp fci^o in tcrtiu: quattuor «ptlnus cnafa illius ^pottionis mt^ nimos.q* IT p^imu boc pacto in omnes Duccre pergis ct fccunou mvltimutquotquot Ubiieni maliqua p:opo:iionc mimmos dfiaes. Cl?IUa8^ottionismtmiBvicenmaterti3tert4repcriltur4rintabtoutoainrei4)ueniatcetairt g«.« btiJueniatoetbinfcefpMnieniatcperfeptimatoctaugfcgi tresnficricoecrfitcontinuejjpoitio fgb————Ii nalesetBDUOoedmStertile^rtremtceruntcotrare^iquareperquarrabui^llittresn&erirunt in 4.<« fua4>potnonemlmi.5teoucoa{noe8mo8etp:oueni3tfgbibincct.pucniatlitcaocroncfgbU fqe-—— runtcotinaejjpotrionalesetejnremiftltcontrarepmtettlltqttuotmruaptopouionemtmi.ecboc *> pjoccfiuquin^fcrfcptgctquotquotvolesrepcriaturtiCoireUriucrmooemoItTationiocognitficft. gb-— - C©inummcoimuc ppoinonalram fuapiopojWonc fucnntmimi ;quotcuqs inter cptrcrooq fuerit mepij touoc inter virffip qctrcfflop rpnuate ppottioales cc puemet bs EjtS-' CSiM«bc» tdkrt c5tfmK pwpdttioiulcs In Tun jpotttone mbni: t>ic« 4itot fuerini tncoQ cSrinoe ptopo4t£ftUa ln< 8<t0menion^rpouionaies int aet totio^ ini davniratl^U qt ,cpibttj i7ui*(IUb<t ^nfierf ptorlmi erfit cSmlTurabiks romomaxioe Mnoa tb(n«e numerates 4 6otef;qiitcrc6mcou«Urifirc6el7ui^erfit(^iitnepu>pouioiule9fruinpniaj^6iUo9miM:tffiop Onoaet mwaimmci&e* 4Sm Utmtqt )? it naan bq» erma RumerSte f et fgm Tnum^oKrnsrwieu uennietnerennciMicFiuBicunivuvHuerwciiomp.exiiegnaeraiorin hetb^fcbin b.qBMte«erso e9ifinbioni eiu« qb dtnutnerfi pabii nuiltipltcw-ivniue in I? «biit eacin a.et (U^ quotie» vnitaa in btenea Uin set 0In o. funt iqif vnitas bea ptinue piopottiSve» f{B’ctvntraab0Ocdttmiepwp(»ti5ale8.4rericDtinlettrenioaatiptuieturbcpuo mcodptopottio* naiea.flbini aet vnttatl ?tinetur ouo ebet intet »et vnitatl itiol ouo 0et bmenil ptopottton^^ qoJ9 eft ptopoiltfi. tntcr vminc^ Duotu ruimiro:^ ermitat£ nummp2opd2tionalc0c()ualed nucro * ccdomnt: mrcr if^os quoc^ totiDe obcrc nccdTocft. Cl^eccScfoeKa pcenldert ini vnitati iacanSt bet eouonSeri cStinue pwpottio^es et ini vni tttl t»caoant b^otiolmeo4 pwpoitlSalea Pico Inf aet etottol capere.*?» qt quotiee vnitaa Inb toties bin e.i5 bin Te ooct^ptooucet etcipl bmfe pwtmcet0et bfboocti in eet 0ptoPitcfit aet D bucoer0obinbetfiatfmanifeltSerftprrptiin3etoctauifcdIel0inptinua pu>po2tt5c elt^bnnb oetnoeoucobinfetfiarbabinfetflatccSftabiripGieptiinSrctfivtaaobitabaocetcapDcaPent itaqtbuo nfimbet cmeDif ptopottiSales tnl 4et pqolaotnooS canebit inter vnitati t a et inter mi» tatenetpufoop ciiptopofltfi* C6^fucnniquotUbctnttineHcotutnueptopo2ttonak0:mimt^quo9DuoctDuo ^9 ,p2opminumcrant:tn caoem p2opo2tionc p20po2non9lc6 crunc. C^tnbcpnomeriptinuea>pottidatesatveromitniquo0 Puocrpuoptot:iminumer3tfl[ntef0 Pico ef0el9tiiuieptcpotrioa(M4'umoenIbbintllap:opom^ mimos tp iq-tertUabc nnmem Cf0lmu.»bcpnfierateorollmb.ei^ocrcptim3^mief0(^tptinucp:opon58le8: qbeptopoiltfi GfUimerod Dtucrfamm p20po2tionum fumerca quoiunt brnid? bints numerati mi la rami: rmtcorranucpzopotnonalcd. GHolofumI nfierosPiuerfappzopojrionfiaquepbinisabinismuni numeratifintp:opoztioaleo PCapioabnfieroscdtrarepmosaqQtoeinaioiiabmmozIquop copontusritc4cummompnon3 tertlilpzim^ct PucobinreetpzoueniatPctbinatpzoueniatectamcerfiatletctnrcet fiatgtper faufmapnmiqpiopoitioaapbealeappctfapctang-capbctccmatnsacrsocnonlfcbim^t ertfapeq>eappet0aofvtcapaqnenonellvtbaoa.nacbeircnteqlesneqivtaaob.naperp2iml bui* qt eflet c5ttnua pmpoitionalicas caP aet aao bpwioecet acStcarent. efl iram ptopottio 0aP f aliaapmpo2qocfaPecteapp.cap(oer0omintmSquenunter3tPe4fitbetmun^q^lnnmer3tef. et ib bet minima quennmerit f0et ftt I.Pico bb1cire cobnuepiopoitioales. nam c14tertii bin e facit b>et qt fao etan^can bet cbfunt^n fna pzopouidcmunier0o ceannem 14 bin fptoaucit b efl er0o bap bvt f ao equarevt caPb.et qt^iPl cPual ia faP ptopuccpfi bnS fefl m!ot et atnaiot er0o eoctana febi erit 18P btan$ caPbJ»bin ffadt Uet cin ffacit l. fun ti0^ Ibbcotinoe pwpot tionalea mimia binis et binis numerofPtuerTara ptopottionfi rumptopnfieraiizqb dlptopolita eSiinter DUO0numeroa quodibet cotmua ptopotrionaUtate rumltuninrcr ytrucn eommct minimu quem omneo numerat toooem aflipon nccciTc cfl. C€totabcpnSeric5ttmiep:<Hioaibalcsqnonct5 tattlcIblnuetuemTmusquInumeratzpico quotnficrl ^timieptopoitibalcs capfit ini act P:»riPlcotinae ptoporndalee caoerc ini aet e et ini pete4lat l0 bmimi qnosab;b cet cPbini etbini numer3t.cr&t l0 bCnoni imius cSrinue in eaPi ,-piopojtioeaaPb«>otti5ak8.3ltlmbnlqDosf0et0bbtntetbtmttumeritf(riiblPUobe«mim5 t* quInomcritabcmJfiefTetmloiBcoireUriamtj tertUfe^eunpInumerarctquaretBiolcotre 4Urtfl bmimus ab eis numerat^anaiotf^lmmoilqb eft lponlbilc.et eopi argiunito left mimosqub -n nomeritbcpetepeoplofipewrmirafiqulmiineratblelminirafiqulnumeritabcp.atmta^qub mimeritabc0potu»eftcl0if cefl mimusquenumerJtUbSint ^erea met nmimipwpoaionis a aPbreraoG t4tmiiamtmeratfetfbetbefmntperip|pnumeratbetbletlercbmm.qaarep octaul rcbl afkelont peittue ptqpoal5alee4t£ tubi e:at9 iiciic ini aet pfunt bcPuo mePli piopoz tionaleszitainteraeteruntPuorbetfntcrpetePoobUtiPlmepUpzopottionale^benptopotitfi CSi pzimud 09 feombu 0ait terttub ao quartu quotcum^ mterpjimu ct fccunoum totibcm inter tertium tquartum ^poraonalitcr meoioe cofiftcrc nccciTc cft. LcS pitf CSbanoplkuteapb:etlnla«PlbubCDuofflePbi:pottt38les.picoenainlef beeouostnePios aipomoales.SumogttuoimimosUUusaipoaioaUtatfiifintblmmB ti fccfipictequapznwitio i0abM. nniiWlb8Pnflcutaappqiwrea<toiteaPb.«q2pcrquintgpfitf6bnruntc8trafepmtcrontDiQ lotb in ina pzopottionc mlm1et bPenmJnonSeiufPe mmimt unnumerabfit equalil et b>numerct tn 4n8Wfnnm«rilmnumcrosfct0ftneunolnumcrfi.etperfcprim3fecunPief0beruntt8rtnuep2opot ——iionaks.ruiti0ilf0itiPlPuotneP4ptoponionalesmle7b^buoPcutlatPuibellpiopoliti{. 1111 15 17 aiiquio cojuro vmmu.9> npamuB numeret vUimu numerabit et feornoO. .wwikVV IWUtIVM* SSlrltS ct> numericoiinue jrpoitwnalcs et an3 numeret b: bko ttullfi coanfieraree. «cmfieft mf« numeratb:c rrta non numerare o.at ^bno nurnemepj^Smim ^ttiomaaao b.pcr equa enCpjopo:twn2 que 4»ponlo eat>g;ea eftb^ju ei « bypotl?ertm.et coAmtlltaj)b9bi0 ano numCTtre oacceptia quattnotmlmia inlUa anxwwone d^nt PUIm«na mm^mue no fit vnit^cahoqut o3anumeraret tIra aoiB)pttro^ -rvltimua uquintibui* CTfit contra fe pilmi %Pt<m^no numerabit m:qaarc neqs anumerabit b.vt mlbao m; ^oequam ®P5 cieoppollto pfine bnfvWmirnoftratl^ en{ ^5nu CSi pamuQ mtemo quoties vnitas in fccunoo: Inter pamu ct tertftim et inter ynU c.ifcfn zfcowiDU numeri ptopomonalcs numero cquales inucnicntur* ?tJjpomonalea Inucntaturtotlbc Inter vnltati tereperiri. nncrabttb.vmraacrffo numeret fquotie0acinb:lteq5fnumcretsqttocte8bnaer*ecttiSeretehi« pertunturmcogpropomonaleavttntcraefotmobtcquoticflpjoporitum. e4fa ga> bUl eSi mcrint ouo numen contra fc pnmi quotcucp inter altcrij eop ct ittinimu quem numcraiiiotiDcm mier rcltquu zvmwfc ptopoznonales reperiri cotmgit. CSlntcotrafcpitmiatcetmlmuaqucniimcratntomicoq-flfnfaet&cabfitbcbno me&flptteue 4)pomonale8toti»e ca»^ Inter vnlratftctqiaeruntcontrtrctnlml.ct{oeo inroaitoottlonemifnl et 0mmunuaqu3numeraBergo qriceflmSluarta teritf anfierat p (erftmm e. ^ergo tottea aIn o quotteavnitaBincergoperpiecepentSquotnumeriinitounnfintertto pifmittmuunaottbem continue ptopottlonalea inuoilanf«iter wtltatcm tc fecunofi: «pwbeft ptopoRtum. CI0ifuerit ouo numeri cotcinicsrqnotcuc^ inter o!rer0 eo? zmanmii <J eoo nila^c tonoe inter rdlqufi zmimm0ab cis numerarii ptinucjjpoztionales ptaterc nceceft» eSintab nfieri ablnulrf om^furabileact cmajrfm^a numeris torn&nus ab eis iuimcrat<*:otco qimt;pp^onal«capfitfnrercetb;tottP?cat>ere interat o.ltf quotcabfitlnrerctastotipecaoere inr bto.Strtt ouetfmtmi in 4>porcione aab b:ergo gricefunatmiatrnli ein cfaelt a^per vice* nmaqu3rr3clurdeinbfaatp:crgopfepnm3rct}iqptopo2tiocaobeae(taaob.ergosbuooecimi bui^quotcfiipcaofitmCTg^oitloaJesinfctbpUmfietrcdmttotlBScaofitinfatoteitlflctquartfi ettterdqtrccabbtiraaaPD.ergopmurattcaoavtbabmquarcgeanp^buobcdntaqaottnepgpM portionales caoei inf cct skorioe casent in/b tc.manifelH eftergo qnoteiiqs inter aiu^ eop et nta pimfi cos numerante abanr.totto£interreltqu5tminimfiab eis numtratficabemqSeftji^i^ eSiab vno piinapio ouo oiomes iiumcro¥j>poaionalra numero cqualt0 fumant qiiotcuncp inter piimum ct altem ejetremo?: totioem inter effremoe mcoii fcciioiun .ppoitionan piopmoium aprimo japoittonalcB eiunt. C^Incipiar ab a ao ovnus orbo numerorfi continue ptoponlonaUS et ab eoo3 aao grecuntws tna< mero quioe cqualigrotco quotmeoR proportionales continebdmrtnter a4o: totibem cortnen inter o t gfm ijpoittone prowmoiaapUmo.cotlneanf ergo bcmeoR ^poittovales mter atbrt efmeoij proportionales infaTgttotioecotineri Picta eflu(cerPg^mp:oportioitlbe:qboflvnbtt.C9pio enf 1; Ir mimos ptoportionisbaP et pueoi? in o^siUos et fwtlmnoeevnapaiterete); altera crcfianc Ipqrper rcptimarecu(U)llinnoruncc9tinaep:opouionalcs:fimilFeclpqr.ctper eanp€qucpro< portiobaoerea eUmaPp.lOeinDeoucaUinae-zfetproneniante tprerunrf^perretttlmarecunois aottrao vricutlaop:papqetqaor>tperoctauameiufDcmnciubabUtitatia08.qa3rcntapp ficuci ap e. et q: pRt cpb Ut *^eUseo per octauam fccunoi paP tvt bao Irutusre vt mao p. runrlginirconttnuepioportion9tesmpt.etqrPict63vt6aPtttaPP:t»UDp:paoqetqaor.cn ergo econuerfo rao qvtvab t: ergo permutatimraovvt qao 1. 1 quU iterum qao pvt tao stergo permutatimqaorvtpaosuiaarepaosni 90 tetrao veruntineaoem proportione, tquia que proo porclomaobeaennaomtoaomtmaolprobataeilvtpaos.ergovtnaDmricaqao tecrgoper; mutatim queproportio naoq:ea efi mao t.crgo per ouooedmam btuuscuminter m1 1 contineatur vnummeolum proportionalefc; pfrainoR proportionem mao p:itioem conrinebif vnum mcoium proportionaleinternqi^ecunoaproporrionfmaopqdntertqrqateet^tnftvci:!: tnfperoctaul primi critpwportioqaovvtl^aolrrquarcTtmaDp.ftuuigtturnjrqvconcUiucproportlonalesfm ois 14 5^ 811 ^l8 4 t b-i-—Lk .+ '^18 I 11 8 _-p m 54 5tf t4 1 io8 71 48 3 rjr 9 b- 10 c50 f45 * 44 btf C9 S1b5 IM «1 pjtrUTftmi cfi<tutjipMtw>oabBOfltrabv»<rgoB«^>wattoat>TvtnaPTcfeotof «cfptfociaatfwjfqTuoitKOi/.jJ^inonajeefmpiopojrbn^fnaPp.crsopertiuotiecuiiiljtilusnB caoocpzopcztloiiftodwmcdtuiAuaair mewltnter ergoper fef^iomecunsi 4ticfpoitio 0ao reacQ DflO g.<rgo per ouocceinti {tuius <iuot mco4pto' pomoiulesconaaunr intero et rfto pzopoulonl maoploHoon contmdiftti^ inter ccr g. mouo »e^T5WfD<»ftraifl«OfmpwpoJtlonimappnBct)4OC«i4ioeineftconttnc£uflittcroctr40Uur «ZBuo castepuipottionj cotumeiuarlater 0eg:<)tS cratoenioa/tranou> eSifJoerttJKUjoilibct namen in lua pKjpojciWKimnimKiJUiatn^ oiiquem lUoaun , * mimcrataitOTtcnnmoniflIuwpzopojnomecnftomcnfurobilJS. ClotniniwffWtitimr 4in allqa^ppoutonefunt mlraUSintergo abcDquotUbet numeri ?^u< jjpottioctaieewrcajipoztvopeiniatnurtfinteftemtmtnayottioncaaobetoianerecgmimcriia 8mibaatctmo:vtvai»c3C9tmscffe<5mea(ur»bileeautfMtptetrc9mm99inj^:tionea^b perfejxabaKi6 4finrbUl4}:tgmiretcoatl4rtoe^ocfcptc<j:eUilfitc4rgopquii^tertflgc<OTtt aicabuaMn«aBtannl.llcmnc«nilUtp»por«um.fiaaicnictsnlujipcriPan cotfclarium cetris fe &Ipcr eaooon qiuuum tertii eomnBieabifewn/:qiuM) cfipMpofitium C0iftjciTOtijuoflibct numen ui fmp^poMioneminimu^pulibct comm ao compo t? fiim 0:nHbqui0mt pnimu$. Cl^nvabcoqootliiKttiBnKricontinoe^oitiosaleeinrasflpottioneiTrinimt.c fverotermtpio poffMi^o a89b.SKoppcaitfi q: abceiTe ptiit^ ao b.lRamfl aliquis numeraret ppofitC e^ta bcec b. 000cpteceMstf«otcabie ome aot fentergo ati48 numerus colrernumei^e aut f. er comporitS c^bc«rs qfitgrqz ergo gsamerat eaut f0ppofltom eje abcet nrnncrfib et vrenp c7fnumerat b etcpei' pfiiuiiaBotdtcrWrfrgoperotceflmatertiSpitmtgnBTneraMtbCTc-ffgoCTgperqafntSfcgi naincrabttarefisufi:fcogp^tas<n0iinumerare 9»ta-coperquOit4t?ulasrumcontrafepilmi ^iturgntmKrattMurafeprimosqbefttmpofibileaidcrgDCopofttusejrabceacdpofuas aotTcb ttccelTartopzbiuianttioDifipzopditiLetuaoequoUbetalioaoeoporaumciereUquidargnmetare 4£Bi quo{ld>ctnimHai ^Knntin fua proportione minimi :qui cjc otnnibua compo^ zo mmraD quemlibet eorumerit prirnud» Gl^ocloeoeQipnraneruscoporitusccouoboaconrrareprimisabqucttbet eo:G per nonam tertOI «npzitflus^cpieeeo^eniquiiibetnumeroptnfuaippomonemiiiimora ao compofitGcxTellquis cllpninaarffgoeopoiituaerillispcrcanci^ nonam anquSibctiUotiim iiumeroiG ctcompofiUic^ rdiquisellpzuausaiuareaoqu^bbetillomm 0itp»inu£:quo9cftp:opontum. eSiftuTiniouo muneri contra fc primi :ternum eis in conanua proportionalitacc ti a^uin^t dltmpolUiiilc. 4n'naairiimtuse(sutcocinoa^po:rionaUtateaoiugerctur:ergopcrp:(m3b<iu>api{musTre4s erantcomenTurabUes: quare no erunt contrafe piini quoDcU contra Ij^otlTcftm 1ptopofUG. CSiouonumcricontrareprimtnumercntalioetmosfmnumcru ancutroconiin la mimcramimittiaciuocp in continua proportionaUtatenon aoiungetur terauo. ^Sintabraimoioontra fe pztml tenumerus neqs ab aneq» ab bnumeratusfmqu4anum00c abnum0ct9:ei{onoBeflenunKrGincontiBuajzpottionaiit^numensc0Da9iuugeneG4Sient fit3iiqiriseftotaefxitperoctauifecGolj)pottlocao9vtaasberlc»rgooaofvtaaDb<ctabrunt «mtrareptimiqnarepCTTlcdimftertflujntlnruajjpottfoBeinfmmt epersedmanonameturoem quotiesanumeratbtoheebmnneratf.rit0gD0tqu3]it numeratfmgxumltat^bnumerctofme zfte&0goperrept>marea>n9iqpM>pottlooaof:eaeaeaogu}uareaqaat>b:eael>eaog.0ab jpbatt funt ta (Ha .p^tttonemUtil.i^rurpoednianonamfediDi abnumerGtegeqoaUf. numerabit igif aUcrnianerop a ont bnumerG e: qnooefi contra brpotirefim. eSi numeri pioportionaU» fticnnt mfua propornonc minimi:non erit cjctrcmi at> if aliquan proportio que primi ao fccunoum. eSinta bc(<ondnoeg>potrioBaI<»inrua^tdoncmtmt:Oicoeat)nuIlGefle^otttonGqaa9b. Si entfu ali4s «noiUeo.qtCTgoaaebfimtcao mergo cmotatla 3« cTicut babb.tqt ab ctunt in rnajjpomonemimiwgocqumtGbm^acfbntcorrafepml. quarepTuefimStert^infua^pottioe fn&niquarep9ecnn3nonIduf9£anGerabttb:tenGerabitt>^ua!natnanGeratb: enabnuerate. nameaoemjipoirio quare pCTvtccnmatertiamptimlTanumerabitctwGtamenacerantcontrafe priim-boc fgirar impofTibilc tptopofitum. CSiquotlibcr numen inruaproporfioncminimilonDcalioanumcrctfm nunieru ^4 aneutro iUmo propornottia numeratu: no erit alma cum illia cotinuc .pportionalie* <tSiRtabcnGert:>{mue 4>po:tion3!eslnfua;ppo:tionem(nimi4numer0tottt>eocf fmUcjlnecp aumeFeFabgneqtabbm^ouioncaaPbm{mis:T>lcoq>oefn6eftaoiGgcnc>raU48culIU8pdnue gipoitionalis^Sl enCfucntaU4sritl4aoiGgaf poflfinGerabnntergoabcrrc&mimitresvUunos ttUusppottiGisef Icqualnfitcp iofmm.tqtcnumcr3CffmUtl^m:B(^ilm3 ergofctfiliaom Cent fao Uqu3rc?f{cutgabl?perPec{m3nonam.ergoccnGab^n{li3^pozdoncminim( uG^abunt bmequalu0uib di coarra{^potl7erim.porttGcntcratneq$gne(7l7 numerare U. un i5 C.Bi fumamur numeri ab vmtatc coanuc pwportionalcsmoUuo co:um numcrabil vliimum nifi fm aliqucmm iIlafmcDifpofito2um» (E^ineabcocftiuniericorinuca^poztioaleeabvnime: MconuUQ eo;: numer9rcfrnU'i^ aliqui in illo oit>ine poUte9.n4mqi quoncovtiitas in &torice e Infrcrgoper olfftnirtonem enumerat ffrfj a.Jrcni qz vt vnltae ac a;ita oa? c.ct-pt aao butac ao f-ergo per equagjpowionalltatcmnvnUaa ao bitaoaof-quoiiceergowitraainbitottcapinf.pcroiffinittondt^fonfieratffctfmb.etqtut^prt' equa ^poznonglttatctnper tlccfunamptiniS fecunpi vt vnitas ao c:ita caof. quoties ergo vnitas in ctoticscinfjquarcciiuiiicratfrcfimfc.etqzmoftramsefloiiumcrareffctftnbzergoaltmutintpcr occauam p:tmt bnumerat ffetim o.ct qt etia crnoUranis cilnlierarc frctim aultematim anfierabic ' frcijme.eilitacpofteiiruinabcocnumcrarefTltun ftn alique tn illo ototeporuo9.q4trat;ppoHtli. is CSinumerojum ab vnitarc conmme pzoportionalm fceunousab vni»tc fucninu mem0pnmua: nullus numerabit vltimu mfi aliquis motoinc anicccocntul. dSfntabconumeripttnue ab vnitate^ipottioaleazeranumerus^uo. bico^nulbisitomerabic i>:nifiali49p2eccocntillolnowincnumcrop3bco.Sieniali({8 altus numeret sfitcqulnfieretefi fcSmf.qtcrgoanueruspmusnucrato^ouctuejecinfwgoBquartSrcrrCanfierat eautf.liter0O vt numeret eintelUgo 5miei nCerosta/t-f/ nerqt per pteceoftem amceptremus fn ectremfi ^mucit o^efioe^oudtcinfmcbiusinmeoia-crgopcrvlceflmaquintamfcSlttaao c:iwfaoc.fj8n8er3t eiergoctfnumerat c.numerctergofnfierficfcSmg:ca»Sroncfvelgniicrflblrb:flttplbfcfij^.Blb€ quocgranoeoduocsgautbniicrar^ atqreanoncnQeruspmuQ.qticptrabppotlivrumatngantb ponaf cqlis atargumetaberiSB pmUTam reliquos ee equales lis qui funt in ototeab Cbiet^^ontfl. „CBumcrioabvmtatcpjopoJtionaUbusfialiquispnmusvlnmumnumcrctirccun^ oum quotp abvnitatc numerabit» (CSint abco 9tinue^ponioales abvnltatetHtip enumerus ptimus quinfierct o.vUimmico i^t e numcrarcafeSmabvnitate.Sienlcptimus nonnOeratatcrgoBptima tcrtfeeritpztmus 39a.et cfi quoties vnitasmatotiesa fit inbiergoafeSmrcnumcrat b^quarepervnoedmatcrt^ben^mi’ 80 c.et cQ quoties Tnitas in atoties bfit in c§ bnSerat cfccSoC a.quare per becimS t^eeft pmus ao c.et ire^ quoties vnitasin atoties ein tcergo cnumerato TedlDum a.pereaoem ergo oedmamze ell pzimus ao o.non ergo enumerat o.quoo eft oppofittzm pofiti. .g C©inumeroJum continue p2opo:tionalium pzimus oc feoinDo ct vlrtmo fi)llaiur t que pjopozno fuent rcRout fccimoi ao pzimn ea eft rdtoni rliimi ao cbpoliium oDn pzcccDcntium» insint a.b.c.o.e.f.ctgk:numcripHRUe4>po:tioalestoetra(^tfabpzimusoeeo(ec5tMznoisr«ri< ODUsfee0oitD.3lteoetrabaFaboeTlr(moglt;rir<preriouusbb.oicoqDeAgoztlotOaoab:ea^eb Nao?poritGepa.b.c.o.etef.rumo entcpgNnfiepgmequaleefttgneqoalec&.ettRteUigogic.gm. ctgR.vttria tot3.qzergognicqu3fefuwpzopoztioglcaoef:eaeiltotiusghaotot6gm.ctqzgn cquaC couoeo que ^poztio c fao coica ell tonus gmao toriz gn.et per ioem q^oztio coao abzea engnaogb<ttvltTa|;pponiototius gkaototugm:eac(lgmoetractigkaognoetranumgm/q' Rumerig.k.c.ftc.opofmtfintpnnuejipoztioalce.ctqjjpoztio toti^gmaorotSgnies^gnoctra crlgmaDgboetracragn.qiiarccquiHtarcdlmUaonmitnmaobnrcriODopaordlouavttotop 30 totaiio eft mIz ao nmificut efao co.ct nmao bRficut coao ab.ergo Bmutatimper tertia fe^oi flciitmlzao efzitanmaococtbnaoabiergoBViccrimamredloibnmUIlrquirfitbizzaobpofltifi epcfcoetabzfieutbnvmius tUop aoabvRutnbo9.atcSgnequerurcoetgbeqneturaiKergogb equibitur cLcquat crgobR nucrolo.ablatisentequalibusabequaltbusirelioua nitequaba.qre ergo quepzopoztio lo ao ab:ca eritb bao ppofitii ep oibns pzeceoltibns.qi} cQ pzopofItuRi. ClPzopoiltis ouobus numeris/fumprifep quotlibct fctfm pMpoztioncm eo? ^Jpozti onalibus:!! minoz in minozein tmaioz in maiorem oucanir tjpouctojum Differentia per Diffcretiamj3pofito^r Diuioaiuric^ibit numerus qmce omnibus alus coaccruaf. CSintabctabouo nunierletbbcopOifferetla:etflntct/ocicf;ctfg:quortibcTin pnnoaa>pozti ORcabaoabtoucatuecpabincDi^ucRiatlm.lfcmabinfSt J^ucniatlnrfltqjoiffercntialmTlo nunzerusom.olco^linmoioioatorBbbioifferltiampumiaorcOm^cnirccgnumeBepcoefg copofltum.inam epabmoefiatloetinefliatlpetinfgGatlq.qzperoedmipztmlqO ntep ouctu abinfgtcquScftciquoofitcpouctuabtbbin efiognGcrfifgietcjcabinfgfitlqietcpa blne<lo2 fit In.crgocpbbinfgpcrcioemocdmaiitnq.TJbicrerca intelUgo quatnioznumeroszabpzimum' abfecitoG cftertidlct fgqaartG.et qz dpzopeztio ptiml ao fefim caeft tcrtG ao quartGtg bvtcdtma quintam fedleiquoo fit epouctu abinfgpzimi in vltlmfiicquQ eft dqui fitepabinefmeoioinmcs Oium.atqiioofitcpablnfgcftlqzcrgoquoofitecablncfcftlq.atJl-oqSfueptetoabmcfcftlp: ergo per oedmam pzinii quoo Gt cp b>> It efeft qp.£teaog pflmtlit? ar^danonciquoo fit cpb b fi‘iOccftpo:ettRcoeftom.quarepernonampzimibbinrDtocgp2ooudtRm.crgoperrcctani peti tioncRi pzimt fi pzeouctuennt oiuioatur per rtoem numerfi bbzrcolbit pzioz numerus cg.qd erat oemonftranoum. CCtuartielemenrozumSritbmcticesJIozoanitznis. ZSO sbb I6tl 01ffermia trium<itwwafa nim ctmtiiiuc caoon I^ I5 ®b a15 b——J4 «ij iu Jtf 9 C- f- 4 *rgfllflin©fl®m<b<aeiIUa:<taprborcrftU9nl*lUo?t»(mfoio9ab0.etvfluincrO0«ju4ti0 blmtbfo. «qMmrMaoinUotf Igtao wtfi {ms.t>ico tr<8 ^ratca numero? r^v/t qttj fint r e:v e:q eJ<i qui pe muriamdi oiTCrftteqoaPraropOnrs reprimam I^Dtue^erPilrer^alaterum in ?polUa e); {^9 jiPucamn^rtiiuCTi&ebi* ej: ©fia ra» V In opofttfi r9q6 eft cj: tfia Vae qin ^pofitum »qqtf eft l mg:eqlOTmimer6j»tidBv(aflm8inquIwmfeclit>tq>eawmntj>po:ttootffcrcfler8&t a» ©fiam aOTq.(Irotaffflre0ariqr8©a^re0atumrv.lTiwiTcntlmga©mg:quooiOcmcft,Sc©l?ocrIrimum numeristeooSingenio inoentia.vtfitarenarius^^ximi ^inct «^minoi:tcmaioncmfi eil bel quinnutcuina ©Fia ao afit o.qua cenfi eft e£ vnitatf nt^e©iffeTfnaDaobTnlrariaa©quinarifi:quaofhirefreqD8temanunLfit(pf©ilFcrcriaba©c^ne qata©upla©incr(ftiebaoa:i&coentbinarfu8.cn(raq$eaof©uplae.re©ob pcrpccptt6enifunta.tt per fcCam pzimi b cfimui funt ouplua ao a.tgifct bcfimobfont ©uplue ab © b.quare que^oitfo e ^^ifrerFneaoMlfermiameadl aggregari bea© aggregatumob.ergoper vicefinwquitamfcbi tm eil quo©ftt ej: ein ©b^mmqtf ex fmbc.crgo per feptimambuiaa tres quaorati numerojB©bx cftmtquaUapiopofmfunrinucniri polTibilcs.quooeftpjopofitum. Cfiumcmm quaoraru inucmrc :<5 di quaoraro oato numerum quaoratu plhruat. 15 dSir^mo ©anianBera9qua©ran]6a:qui©iutUbtUer(tinBuoequalia.culu9me©ieta8ntb>latua ofit etua fit t.flfq5 ©binariu0.quia ergo q© (ieex ©mbtequaf ei quo©fit ex cmfcwrit BvicefimIfcxtB rec&oi c4>ponfoaliter meoius inter binarium xb.et quia ©binariua eft numerus ptimus: xBpzimi quarti cSicatcfietc <8 b:rtiaec3icabitffib^jj5m3§remj©nfierabitb.rtJius4oebmie©ief8s fit e» «n6<ricircfipofitifintfmaia:xsratnot:quoBq©nuafintb*lt.oico5quaoratfiI?e2i3nq>nnmer& quaoranim U:numero quaorato aa©©irum.Wam©Ilferetia fxgcbmarius xppofitus ex (pfis per rmoamp2imie(tb.et quia ex binarioinb per i^yporbefim tu aiergo per feptimam l^uiuscfiai^tex ©Jn «tftia (areri in copofitC ex ipfls aefi ©fia 4©raro? btU.^orat*’ ergo Ua©©itus quaorato©aro a pihnurnunterum quaeratum b^©eft jjpofltum.SCT eftofccfioo vt fit qoaoratue aaffignatustfeo InOuo equalia pamrtnopoflit.cii ergo binarius eum no numerenet fir binarius numerus primus: ergo per primamtertii binarius a© tpmcrit primue.quareE ©cdmaquintam terrO binari’ nfierabit inferiorem numerum quaorato a.*vnitafc ©empta qui fttb.cutus meoierasfit «x maioteovnicatefit ©unius qua©ratus<ttc:etquaDrac^cfitf.©lco ergo qua©r3rume8b8©8refuBqua©ratoa9’tum cfi fquaoratosc.Wam quia cbts cfi tnitatc pfiitoit aiergo per ©edmaoctauam primi quoo fit ex cinfe cfi eo q© fitexain ©fiamomsioris partisa©e.'cqu5cfieiq©fitex©ma(orcparrcinfc.excc©itigirur quaoratuso quaeratum nucri c in quaorato ©ato a.nam ©iffereria ©xcerat vniras.q© cfi ^pofitfi, CSi qu3DratU0 mquaoratum oucaturJjJDuccf quaPrarus.Si ^0 in non quoDratq 14 non quaorawsjjucniet.^noc P5 1! quaoraws quaoranim numcrctupm fccunoum quaorarum numerare. CSlntMoaxb©uoquaorati:©ucatur(S8inbeta5Heniate.oifoceequaoratC.1flamB correlariS quaiTebttiusincaetbeftvnus^tpoitioalifmeotus.flt ergo «le ©.quia crgoaoxbffitprtnuejppot «oal£8:cr®BWraimafextamfecanoiquoefitex8mb:«iuficfteiquifitex©mfe.atquo©ex ainb joudtureuc4dt igiturcqiuoratus-q© inteoitor.^ecfiooifitaqnaoratusxbnonquaoratusibuca “r*B «w» b«ameniat emico cnoneffe qua©ratam.*amfi ceflet quaoratuserc©am Btcm©tcefime fexie feainMlanis qoaorati cdtmeotus proportioalis inter axb.quare cfi aprimus fit quaoratus Boedmi(joioatbeftqoratus.q© eft xtr8l?yp«tl?efim.lCorrelarifiexl?teoaabusBrtb!> fatis prtat. Ct^uaoratus non quaoratumtfrtm non quaDratum numerat, s» CBamfiponafpoffibilequaoratfifcfimquaoratfiilpm numerarciergopcr primam Btcm ^©ftf gmt^^tqMo^r^s^tpM^ tloratuanumerarenoqusorammifc^tur igitur exboeptrartfi f= C.Qi cubuo moitnim oucaturiqm piooueetur en’t cubud, CSfntabooocBbirxoueatorainbctjentatcoicoceflccubum.SumocrtnatuscubiaquoOfit ©«tquaoratumIaterl6©4fite.manifeftumeftoxcc©proportt58le8 inter vnitatem et 8.etqraties ©nirasinatoticebtncfcfiamuttiplicetcrcfimb.peroedmaquartainquarttquotmeolljmottdales erfitintervnft8temxartori©emcrfitintcrbetc.fnntergoouome©i|rmrerproportioalesbtfbetcd fuitfg.cfi ergo bfgcfUit quattuor numerijjportioalesiet bprimus per bfpotbcfim fit cubus:erao pervnocdmam ijuius xcquartus erit cubu8.quo© eft propofitum. eSiin non cubum cubua oucaturtpjooucctur no cubue. £j: quo liquet 9fi cubuo n cubum numercwpm febm cubum numerare. educatura cubus in numemmbt» cubum ctjmcniatcrolco cnon efie cubum. iHam fi c^©uctus ponatureicubusrmanifcftumeftvtirtpreceoentiperocdmamquartamquartiquotmeoii ^iponioa leserfitinrerTnitatemctarrotioemefleintcrbxc.firttergofimiUterouorciljfxg.quiaergoegfb ftint ptinueproporflonalcstx ccfi pofttus cubus-ergo permoedmam buiueibetiam erit cubus, at bpcrbypotbeflmporitU8enoncnbuB:acdoctira<b?tr3dii(?rp©tbrf‘9-qunsiturin©C4>oucctunn6 erti cubus^iuo© eftpropofitum.iCoirclarium faris cx bac <piceeoenti cognitum eft. eSinon cubum cnbuo numa'ct:fa5m non cubum cuiti numerabit. ,s Vi ponaf numeraH %cubfi:s penoltiinj «loi 4>t>ucetur erit <ut>u8« at p«rittt9mn? cubus uo Uacp iprum numerabit ^mcubgicjuarcfli^Bm cubus numeret:^ uo cub&nScrabmqbtntfuitur. 19 c0i qu4D:an' comuiticantiatera qao(5 c6inunicabunt.fi woquanaatl fintincoms incnrurabilc0:ct latera ipfoJum incomcnfurabilia erunt. CCSincabouoquaoiaactcUtusaetsUmsbioUopiimoftctuaoiatiabcSicanteteoiaUteracu coicarc.na ft cono c^ioinnergo per ouoDcdml rertd necp quaoiari abeSUabunt. ct oppoflto igitur pntiaflquaoiatlc&cabuntfeoplatera c5tcant.oicorct}o fi quat>tat(abnoncdlcaninc<peo9Utera c6icarc4)ificolaterac5icantdl8«<}9nfieru0eO9coitcrnumeras<lllteu?:<rgoenumcratccrgoE vkcfimaierttaptlinietta numerat a.et AteriaetiaeratoiBcanbeeiccfimaterttlnfierabitetb.ficrio later3funtc6muntc5tlaetqu90tarlc5lc3nt:quareej:oppontopfttt8(i<iuaD:atitwncomunicitn<fl9 Utera coniunlabant:quoo dtfecunofl ttotum ptopofimm. "> eSicubi fuerit pmSfurabilcoi bfa.? II cubi no coiccntiuc® ipfo? Utera coicabut CIbuiusptima pars vt ptecebcntis pUma-t fcctuiba vr ptecevenrUfircanPa pacefeont. ti C0i quaotati aD aliquem fiicnt ptopotQb tan$ quaotati ao quaotamm lUum aii* qucm:quao:aiuin ciTc ncccffc cft. GSitaquaotamequiaualiquenumcr&vtbeambabrat^pontonSqnequaDtaricabquabUtfib tticobeirenamcrfiquaotacij.naqtcetoruntquaotatipercotrelariSquartebuiuseUTOus meoius ^pozn’onalUintercetoquirite.f qt que^pouiocaubca eftaaobiergo per ODooedtnaquarti erit t>nu8meo(nsptopottiona>i8mteraetbqmritf.qter0oafbrunt jmotrioRalestaptitn^pofitusi qua&:atu8:ergoperocdmal?uio8etbtertlue eltquactatusiquoseupiopoTitum» (E^utcuiiq; fe I75 ao cubu ficut cubud ao cubu: ibc eje nccefiitatc erit cubud* (Calati anumerS cubum Te babeatalEquUuumemevtbftcuto cubus ab cctibSn^icobeire cubum na per cotreUriQ quarte butue inter cer oerunt Duo meod ^poitionales.et cfi que 4>pcutio cao Dea fit aab b.ergo per DuoDecimS quarti cacent cdflmtltter duo meoli portionales inter aet b:et aprb mus ponitur cubus erit igitur^b per.vnocdma buius cubus quoD erat DemS^ranDum» C0iqu3D2atu0 mquaojatum.vcl cubue mcubum Ducaturilaius pzooucti erit ttus merus qut or latere i^mus in lama altcnua pJODucitur. CSintpifmoDuo quactatia b:et Urus afitcietlMsbfttD.DucofqtaiRlittrpcniate^per Decidi maquarta bulos erit quaoratusietDucafcin D:t peniatf.Dkofeflctcffagontcft kttmtMiimftp equiiefiet^fitepfinfe.ilelfflatue^ua^atie<f6eratJ;ibaito6JefofutB«tQ^)iiHUutiltefli.tg' inbpucatltc{pa’DedmarcFt9bu(useritcubue.DicofeQe(atDSCOb^^«mIcUlapitti!ckecf infe^Dudreeritpnona quinti portio gac ecopofltaacpDrti<Mt^caDretaaDf.ctqritcrSei:e inDntf:crct©lnbfitbmtpcrcanDlnoniponiof8Dbt®P®*^^f>^a>P®*ti**‘fl>^ctDDtt©.aobtfj caDOcnricutaaDf:reDetpermutatfoaDbricutcaof:qfeaD2efret3i^tIiocaDDetfaDb.entgltar portio faubequalis pouionibusaaDfetcaD^quareporttonigaDt.oroinatiee^quartUot n&erlsgcfbiqt quepouiDgabeeaenfaDb^crgogvicefimaqulnnrecunDiioicftniKrusjpil dturet;ginf?eee]cfmeru6quiDeqDaD:atQquarepuaicUbiU[atuslfiqbefttot0propofit4. i4 €01 Dirpofi tia ab vnitate numena quaD2atia:^po2tionaUtcr mcb^ inter ptotamo^ quofe^ fumantunfient omnea parte altera ldn^02ca.ct ouufitbct lUozum aotUoa « Differentiam latcrifuocqualcmdTcconftabit. C<5lntefgbquotlibet nfieri quabati abvnitate pfiter DltpoUttieopvero lateri vninldtet bcDn» meri pnrer abvnitate rumpti.Dico meotos^portionales int pwdmosquofqt quacratos fumptos ei. altera pre logfores.Duco enl vnitate in btpeniat Ir 4eil merfuspornonalis tnter e«ftet q: eiu4’ lateravnitas et bfont{equaltarolavnitateDill3tia:B oiifinltion£lreft^erastelqncier.fiirDucob incet^ueniatl4eritmeDtU8poitidalislnfprortmo6quac^oef0.etperio2vtprius altera Qte longior.etitaDercIt^sporrtonaliD^meDUsprodmopquaDratorSquotquotpnter aiTignatifuerit quaDtatUl3icoprecere3Differenti3UabquaDiatofequaleeirebUterif»tDilfcrentiilaquaDUtog ciTecqualecIaterigtetltacorcquenter.qrenibruperatvnicatefoUimitateiinteUigoergobDiuifum in Duas vntrares.et qib in fe facit fiergoper Dedmatertiam prirntb Ductusin prlm3 «feconoam vni tatem procrearequ&^feDfub(h'actoq4nte):binvnavniratercmanetb.elltgitur&ibflractfieiusq^. fi tex bin vnavnitote Dtiferfria Ir ac fifeo et iDcm rubOractfi eO equale b:^ vnitwin qu&utu^nu^. rumDuctaisemfactat.connat ergo iD dTeverumbelr.De lautem confimiliter oftenDitunqr ccteoib bfola vnitate.DiuiDo ergo clatus fcilicet quaorati gin bemittatem.i quia c in fe facitget toei^ in betvnitarcmfadtequalegtpereanDemDedmStertiam primnetrubib-acroeoqulfitcxclnvtuAtciR remaneri. Differentia iglturletgenquoDftcexcinvnitatan:teotUluDe{lcqualec per loem q<leb ptlus.conllatlgiturpioporuumoctiro-aparrelongiorel.ctitaDeqnlbuflibetaUis argnmentabeit Similiter etiam oflenDee Differentiam Uao eefTevnitatemiet Differentiam lao f cfTe bet maDg cflb c. lRatn cum bfola vnitate fuperet vnttatetndntelleeto ergo bin onasviUtaies Diotro per occinUm* quartam primiquoD fit exbin alteram vnitatumeqoaledlet qui iit ex iUafnfcttUU in alteratn.r<9 1e1? <1$fte UU'mftatefn re eaqoaototfi e t qtf Rt o: iUa mdl'mftas et equilrmi ntUonaster^o DtUtnhsoeeOeniitai^^SimiUter &(utt>o c in otuspamsfovnitari tb:et qt per esnnl occcitiSquartam ptimi c^o Rt ep cin bUcquScAdqutRtet; bin fe 4rAqiubutiia fiicfb invnitarem qui cAUtos b.(onRst ergo blffcrentii Uofefreb;et in per ncdmiquartampdmibe iTn gDUeoaenbe.conftat ergototumptoponmm. C0irumanatr ab vmtatc nummatecra paitc longiojca pjotimia quibuf® «rniun « ctie: oimioiia comunctojum fumptiB pjouaticntomnca quaojati CBtparte aiteta iogiotOma Rguratia qoSaomobG in rupiotl factS ^otco Uet laltera IdgiotS (Irnt fumptoprnebimtl mequabnt6>SimlUIletm Omnifumptop mebierate eife quanata aita binop etblno:5pdflaeriunpto?quorquote(renc;fflenietat2efl*etetragonid'. iRam qaper reamti partem (ccocntl«nionarataeao:Umkinino2t0aofeireb»tlnuiod8abfeti3e(reb:liairuntcircfipolUf at(9 cquioiligtea f.ergo qfecfioSpiimi feft numerop bet lfimul mptotQ mebietaant fcognofeif ei quabmtS:<i^o beli et 1conlhtt 4>po(lnim efe verO^cSAmiliter beIamcoftabit: qt eopbUFerentia ab gper ^ceemte eAclamagtquareperfccunoR pnmlcondnblmr quaozatua gefle numerop 1et m flm^iunctowmmeoientem. Cl^ullud altera parte Ion0io2dt quaDtatuavdaibua* ^ Aauqa altera cte longiot vt aefTet quab:at«:per oiRinitioni eina latera efTent equalla. et qa metalteraBtelongioietualarcraroUivnltatebiftarttdlcrgotllab&aretflntbmaiottcminottqa e^aeiTetquabtatrrtteiuataragonldllatueb.etqtqqRteebineeaequaleeiqRteiebinretergo CvicefimafeeU reennoi bbcfunt cSiinuespottionalea q6 eft ^lofllbile.Tlam ctun bet cfola vnitate biOft inter btcnullus eabit tUiems mebr.reb bbteat abuerOiri* be2 equaie baut c.ca ergo bcocb tinne4>ponionHopotfetbetccireequaliaqbeacdtral?rpotbcrim.3tbicatbeiretnaio:cbtc.tik ergo bab beA 4)poztio mtone nfieriab niaio:e:a bab cgipottiomalorf ab minoze.et illeponfif eebe ^ocaRt^ofnbiie«fitblcstvitbbTtro<petbaccfreininotg:ectficg>pottiobaoeemaiorr8bminozc ctbabcmionaabmaioz&tpononFeetbl^ocitqtlpoinbilcnocAigiCaalteraetclogionquabutua qbeAptiro6>oA£bobeinbeaalteraBtclong(o:gn6e(recnb{Unanibpo(ribile2rAaaltetastelogioe cubas teiaslafanntmiaiuaqDibebctminuecetqflabnrfibritbfqoabntaeritetAttbflaiusain eo^cubammi^quabxatfiAt U.crltenIccotrelarifi quarte bui*vnusitiebtus;pponi5aiL6lntbetk qAtg.etcib£vnu8inf e‘tUqAt>7:eBrcptim3reambiq;ppozriobaofeacAbabg:etgabb.etri{y q^ottiocabfcaertte3bbetl73bU.etq2eeU(nfRteeibeRtpt?rporl2enmct:bme:ergopfcba; ^rtgTicerimequberecfioiertrhabbricutcaof.aetiicmutarnz abcricutbabf.crfitig(tinterbac cStiiweMomonsIrsgetkeriAinterezb.ltlbcm buo bctlt.etq2bTeve[erunt^m^urabiles vel eStra fepini.fi pmdbrablles eSbnumeret bems qaab2at&:ergo pvUefimSrertlam pzimi qcuncp nu merus n&erabitbetianumerabitb.quarc btecruntcomcnrurabiles.at tft qtbetcrunteotrafepmi beteeirecomenfurabilesBbuooectmiterti|eACpofnbiIe.n aStbrerintincomenrarabilesBrcbam ^ui*bcrcfuntcubi.Sirrfic«oeirentc3menfurablle0Blocni qb piiusbetcefTentpmenfurabiles qggouobecim3terri|cAipoinbile.Sia6tcabfinttncomefur8bllceergoBfcbaml?iu’vtcr{plcub’ quareinf bet cerfit boo mebia ptinuejipoztioaUa gcotrelarifi quarte bai<' at no mo buo vep etvnS qntb2 mfttber€inebl6vtinpmep8rtem6Ar8t6eA;cAIpoinbll«.c5Aat Igitur verfiefle^ipofitum. CBato altera parte lonstotc:altum ab co altera parte longiotcm inuenire aiiua ao 17 Illum fu piopottlo que quaozatt ao quaPiatom. eSitabatusnSemsaltera Bte longietet eius maius latus fit bnnln^^ofltcifitcp bbuptus ab b tebuplnsabcnquabnt^britftet^mnsAib eo quabtat^fitgqno nfierusvnitatemtointl^tBte ^teralo^inter f et gCit k«qt cni per fecunba parti viceTimequarte l^uiue bzla fct Iz eAb:et bzia Iz tb geAmSems ee minozvnitate.ibeo bzU bab bIequalis b.et cfi bdabet e At binanus: qiR buplu bdebab c.ni per eotrelaiifi fepttme febi buplatop:btmlerunt ozie.t c6 ep bin fe flat fet ep oin bina: .ria qui cAbzia eius ab eRat bdafab b:qi flt bzla fabketltabhqeAbupla ab bnrt ^oAefum eA.erit preliquus nfierua qRt epbin e.buco igtf gin pqnumeri lolavnitate oiAare font poruiiRatcp l flteractelongiotqulbicoefre:qalqnerim9*bucaCenib(npetR3tm:tcing7Ratn.erlt<;pBoctau3 tcc^ea ppomolabmqgab b>et per eanbl etia ea^oztio nabaque gao b^uareeaeA ^ouiol ab mquenaoa^aep bin pRt m:et epein gRr nnbeo cnoni qninti eritjipoztio mao nppofite ep 4)poztionams babcetpab g.p qab in re Aet0in efadtperit 4>pozno f30 pq0 ab eguere tq babcetqajipoztie faogcSpofindtep ;ppozrlonlbu8 fab pequali baoe et pab giergo^oztio m ab neqnaf.ppozttoni babg^4aper cozrdariii quarte pnia inter fet geAvnue ^ipstnonalF nicbi* erit vnus Amilll ;ppoznonaUfmebiosinter met nqAt0^Atergo 0 abnHeutmab 0et n ab aTim lao m.ergo per equi;ppoztionaUt8tlinbirecr3 erit tao 0Acut 0 ao a. capio ergo injipoz tione tao o nesnumeros in cotinua,|)(NtttionaUtatemuno8qAntrqp:m9nifcAfieftretpeAequ8bnros:runc Acqgjpoztio rao qca II ab otet qqab pee eA 0 ao aiergo bequa ^ipoztioaUrate que jipoztlo rao p qaa^tiaoquaozatfieaeAlaoaaiterapartelongtotisabalterapattelongiozIrqb eA;ppoAta« pfSl tiaram ab vnitntt fpatttonalcs oirponalur tmtora ab vnitate quaoiatuiatt® >* vno Intcrmiffo tertium fempaefle quaoratu quartti vero ab vratate tubum rtmob’ ifanniflio quartu Temper ^ecubnmtrcptirau autem quaomtu culutum ati; quinip reptimu femper cITc quabiaium cubicum efl nccefle 1? JO 31 31' 35 VI ' £fiiuozsru«CQblcu0enqaa9»uu94pArit(reftetcubu6*‘^ontaenlvnitatetbcoefgbUImn ^ 0p:vcl quotquotvoles nnmerts cotimieab vnicatc^tpottionaUbus iTtenanm ecocairu tvnoccona bum0CoaJiot«Mtc«i'?n«4*p<«a»etquao:«t’cll'rcubuo.i«opero«anul>mu«ceniqu4O«t* £_ <tcvn»ccnnascrUcubU6.tp«reanoe»ecmu€Ucrumerit quaoiarusetuerfipcroecimStqieeft quaotatua smtquaeiar*.Sco ct q: ocft cubne pvnoectnii getia ertr cubus 4vernt vt uqoo coftat jj_ repnniuoab vnirateuiuarerepritnus ab vnitatc quaoiatus ecublais.et lU edeinue pioceoe ftmU^ onenoes Demoiid^not mooum vbtq3< CSi numeroj02b Tniutc c5nnucj>po:tionaUu fccuouc ab vnitatc fuerit quaotat* omneo cruniqoaDtan. q» fi iocm fuent cubus: erunt ct omnes cubi. C;'Bb:ima pars patet.na per pinifilctla terti’ erit quaoiatus.t t{afecSou6 cftquaotatua cberim3 ^utus ternos ab lUo ij eft quartus mani eUquaozatus.et per prailTam 4a tertios eftqnao»cusvno tntcrmilTo qititos mt quaii:atu6.ei tta oeitteeps quarenumero? ab vnimecSrtmiejjpottionalium fl i^e^ouo e(l quaotatus oes erunt quaotari* Sebapars monrator. ni flfeefiDus fuertreobustcom quoricsvnuas in fecfioo totus fecfious in tertio-febus tn Tc 4eft cubus pojoucet tcmfiiquare p»eci inamfct:ta l^uius ternuserit cubusut per ^mitTam quartus etil erit <ubus.et pervnoecunibnius a quolibetillotfi aiTumprusquartusertt cubus.atiUiscbttRuerunipasoesrumentur.connatitaq}fl fecfious ^rnro^us omne» cubos efle. CSiptoiainom altera parte longio? vnus in aliu oucaft^>ucniet altera pte longio: <CSint act bcuo bunc in moofi alura parte logiotes et raeotus tnter eosquaezatus fit ccui» Urus ritbqbpcrvic^imaquartaf?ufu8 erit otfferbtia inter aete Tiniercetb.fitqja minotio altera parte longionslatusaltcrfic.crittpomaiusevnttate.i qt oin fe facit ceroin efacitaerit^ottiocaoft tan&oaoe.Situb vt ccoinffiacbentfvnitaremaioio.aoDaf itat^caoaetfitcoponfusgzqteiro In c Mact fin eftet gtvt facilep occima piimi oftdoutur cu numerus fnumerfio foUvnitate ftiperct etqafmcfatitgcioincfaatateritpcroctauafecunoiet jjpottiogataqcfifaoo.icoiEfcpiimJ ciufbceti3baDcficotfaDo:quarc«ba»cflcurg8Pa.quareBmut9rfbaPgfiaitca!>a:frgoevirefimaqmntafecunolqbfitctblnaoatlsaltcrapartelogioiiboseqnaleefteiijfitetsmcatqfitctg tnceft altcrapte tongio:‘B»iffinition€:cegetcrola vnitatc oiffcrat.q-giitcopofims^taeteTonu 8ab Ipfo quawato cfit oia qua 4emtuta onumerus caoottus aincopoflnonefola oiftat mltatc. coilai Igitur mimerfi qui 4)ouccrur e* bin aefle altera parte longlotb.t ptopoficum. eSii>wnu quicp pte altera !6gio:cs cu ouplo quaetati inter cos cofiiturt pponanf ^cnictquaotat’ cui” raoi? ent eje minotemiotis zmatote maiotis latcnb^pftituta eSintent aetboooqafoSqsptotimtalteraccelongioresereo:fimesiusquaozatitfc.f{nt90cU« tera ^erapte ISgiotis att eflatera alfins altera pte longtotis b.oieo q> il a b cSponiiarcfi sopio c totfi aggregato «geffe quafiatfiOTiios rabfjr eft cSpollra ey oet f.na p^cefi in3qaarr3 bul«s eeO o:{3aai>eetbabc:ergoprecanbIptim(eSponmmera?benMiplfiaoe.atboplficai>»abbtiplo ctuib^ erunt eius qnaomptfi qcoo4bcm pofitfi eft g.qoattuot igif in c^o^t gwit quattootniiems ellquaviaruacum piooucafecouobusinfebactis-tfntcpolbuseftqnaputus.ergopoecftajquar tam pfcnnscrgcftquaozaras.etcfigqaaoiarosquaicupl^ntabcqoaoiatfi: ergo per terna butus Bpoztlolafi8gaoUtu8ceritoupla.atdief{trabircet»Tfcircflpofttle4oiftoiri|Wc.eigopr(n5a5 pzimt e etffunulfuntcnplfl elateris «quarcrabitquaoutigcftpporitaetoctfzqfi efl^fufi. buo quiUbet ptojnmi quaotan' cu ouplo rncoq mter cos altera pte longiozisco iunffatuncopoficus cntquao:aiu9 latufcp tpfius cr latcnb” ipfor compofitO. esintabouo quilibet pzoyinuquaozatiet cmeoiuslntereos altertpartciogtotctoeeoBlatenr Pico quaozatos abcfi ouplo cefficere quaozatfivt fent’ Unis erit copoflmmeto e.iftam pvtccfbna quartfbuiuseroiijefiefcaltcrapartclogioz inter aet bzquare per nona ptfmicninracrctanq-qb fitetotnppofitfieroe:quo»4oem copofirumfitgctpereanoJccfibtan^^quoo fiicteinppofttom oeio cfling.etpcanoinonSqdcyocteingtanq-queolitetgmft.eritigifquaozatfimimcngrfi &tum aTctcbzergo ^tum aet b«cbts.fulr eni cbis fo femdcfi act femcl cotn bfumptfizfco illuo quoo fiebat et act bicbls pofitfi eft effcf-ell igitur fquaotatus eoius Uras eft g. et 4agcnppefit* vt pofitum eft c); 0et elatcrtbua quaotatozfi aet b: conflat totom pzepofitum. eSi90 DUO0 vfep nucroo vnitate otfiatcs: oes numaiab vmtatccoaccrucmr:(j ct coaccruanonc ptoucnict erit nucrus quaotatus btufq? ipfius coaccr uato?: manm”. ITSint nfieriabvnitatcoifpofttip tuos o:olesvf<itaoctoouos4»cnficro0vnitatcotftates^mus bocmfivnltasajjcor.tfcbasbocmovnitasabcoitfuquaozafenumerusf.oicequaoittfifrirc tanS 0^6 nomeriillorfiooo? ozoinficoaecruati Signo enl oes quaozaios ifeiiozes f4fint oifpofiti Dezotn«fgbl*lvnft86.erit(»gquaoiatvowtbqna»«f witaeqeftpmevntt8tfsquaoi«’.3nteUfgoftfrmnopqmeoieoaltcraBtcl5gioze8.et4a pervice fimlqo8rf3bora®w*af*®”’«i'^®*^8maegcfto. ergoBoetauaccnccptioneoitaettrfmopfgb eo.« ftn argumeto oila gao b^1®wiabaoUeft cb:« etUUao leftb awtozia lao vnitatc eft a etvnttas.aoofta crgovnfiate ao attvnitat£fi«l.et aooltis baozia fir3'ao U.80 atoupUvmtafe.i. aol:fictl2.aooitainfuECb©zIalz80bBObaTaeteapl5vnifatc.i.aolzfiet bi^fimilFaotitaocoiia baogaocbibazaetouasvmtatesiboceftaobtffctg.ctaooitacoozlag ao fao occb.b a;a ct oua6vnitate8fietfq>er nonam igitur conceprioni coitftaf ^fttom ocmonftraram. 5 -i5tf —5 u -is+ 719 ^$66914^9 .£-0 145 4J48907 bSi y,- 5904» r7 19'S85 b 9 •l• 7X9^ e 3 ni— 17 fn—- it g t a—— _,f «EgSracuncB cf niimcrte ab vntotc bie fumptia coaccruatur:cO parte altera longioa mmufCB laterum ipfiua^oaceniaiojfimapmua ... CStncbuiafmooinecribuio ominefiinipn wiltafl abcc4flmol aggrcgtwr fiat* m: auefico rf #l«ra Et« U>n8i«nc*.daf<? mma«Um« e< c-ni p«r pagentifi poictie nfima atoo«fiet qua^tama fx ailiia Dtia amewaaltera Pte l5filoie cft e.rcnioto tgif numero caquaszato fremanet altera Bte Ion gl<M meoi’ iirf f«pjopmfimtnoji quanuASeo4remanet amoto eeft aggregat^ oaopafngnato? aIteraptel^oifiter»lneiUotaqaawato?Utenb*peiTUcflmaquartapftn6 0aatminu6eifcergo altera EJeWalorfa mmlnrlam» iti.at oemtnfiafiignatop majnm^ enfierue.^lUt gtota 9Po firma CSifucritnucrusSjoiatuBCubtc^ildtusuuiDcqoialicrilcub^Iatue^oaibiqiHat® 5^ esitanfierua quaojat’ cnblca» tni» qnarem»qoaowttlamafltb:quawuevero cnbi latus fit c: ©Icobeirecuba5ctc^equaoiatfi.^^tlmfip5.ouco cnlbinaTfiatOBP^l^^oefferobnetaaibus Ipmnumeratfm b:cTgD peotrdariffeecimefepttmeljutus bdt cabus qS eftpzlmB.Sc^i oRftjJtur nteqoao:at&cqi qdfubcetemrCTiiscfitlnctureqaflefteiqfi fitetblnfc.erMBvtccrimafeiwm rccuRDl ebeftjntcontlnoepmpozrtonaJesfeocpzlmuseftpolUttsqoawatus.' Igitur cicpcroeama i?uius ellquautatus^cq; totum conflat piopofitum. CSifuennt ouo immcri qitaowrt cubid :meoiuB inter illoe in continua ptopottio* nalitatc ent cubustouojum vero mcoioifi vicri^ quaoratue. G^nt aetbooo c^oiattcubldtqtatbfuntquawatlBcotrelarlfiquarte Oufosilntereosvnu* meoiusjipoaionalis dflt c. nqt aebfunt cublp toem cotrelanfi inter eottei mnt ouo mcoq ^pot< nonalc6qfinto<e.olcoergocCTecnb6etete^equao:ato8.c8fitefrccnb6occlaraf. naruoucea in bet suentat ftqtaibfuni cabltferiaperoecimafepam Ijuius erttcubusetcuacibnntptopots tlonalestergo bvlceflm8fertam fecfioifcubus fiet e):< In fe. quarefellquwzatus Ipfias c. eit^tur fquawai* cubicoszquarcc latus fvt quaotatl per pccoenti ril cubos qfi eft pumfi-feo oet eeirt qua matosbine Uquet: qi be»funt cotimie putpottionales er bptimus efl quawatus. tgitur ttertius o per oecimi bolos. 3tem at> et efuot numeri 4>po«ionalc8 et a ptimus ell quautatus iglf per eanoe occimi teremos qaamatus.£Q Igttur tota^opofitio nota. C. Buo0qo:ato0 cubicos iucnircnnf quoe nucri quotlibct mco^' ^potooalf fumaC 37 GSiuoosqojdtoscubicosiubcoifuenireinferquos quotUbctln?iacc3tmeoB ^potrionalesTumo nfiero8.ppo:ttotialC8 ples ouobuszipfis meoiisiquos lubeoi iuenire.verfat c3 f» tubcot luenlre ouos qu3o:aro8 cubicos Inf quos finr quattuot meofi ^jporttonalee fumo fepnfieros cotinue^ppottioales vtabcoeftquozB perozoiflSfomarurcubigbltlmniqpcrfeyrSbutusfuntctiajjpoirtonalesqme qoio^cubimfeperotomfoDCImrefptoueniatopqrsr.oico tta<poct tefleouoo tales quamatos cubtcos.nacomgbetceteriinfeoucaturopq‘r«terifuntquao:a«;etq:gb'tceterteo:u fcj late rafnnt^otnoalesq>er quintS b^liiA 0pqet ceteri eotfi quamati funt ^ipottionales.at cum get n flnt cubi ergo per occimlfepdbuiu6oetteruntcubl.funttgtturoet tquamatt cubid imer quos quartuo: numeri pqrsfunt meoi| ptopottionalestqfi e(l ptopofUfi. G©uo0^o:atO0cubicc0rcQirc:m6 ^0 tticoioofmoatoonucroofum^fitpolTibilc GSintbatiRfieriatbiiiporicficpfU ouosquamatoscublcostuenlrctinterquosnntnumerijipot 5S nonales fim atetia ^btfttqt abinarius et btemari^.cia entjipotttoncs femp funtvnop^res tpfls meoiisetaeflbinanusuSvtmeofltnueniaturfmaopouettrcscfleivottfoncs.ct^abefltcnjarl* wInueni3tormcoiffcfimbop3 quattuotcffe^ipoitioncs.^uco ergo tria in quattuoret gtoueniatc etfcfimcotolnocdtlnue jjpomoestnnOerisrpefgbUlmnopqiquopqetrBpccbenreponantur •e quamaticubidretplfigoquatniorepilliSiiportloib^vrbicfcSmquafrtosetqternosomestabtbs •p tresmcPios.ppoittSaTesfcjnUfBequS^ortlgaUtatl.tfiEl’ plungotreacjrUUsvtbiccqui”^* r^nosowlneeetbftom-iguuosmePlosijporrtoalesiniqetrquaotatqscubicestqBcfl^pefirii CW4mbct 0U09 fupftdalra niits-.tcmu in ppnuaOTOJtioblilatc mcoiil cc nece cft CSlntaetbnfierifuBfidalesftttsterlateraaflntcaoietlaferabCntetf.PlcolntaetbeJvniinu —merfitttc5ttnoa4>potttonaUtate.n84aactbruntrtre8B Plffiuttion^rius latera funt gipottionallii. ioeo fi jpponio cauo»a dlcao f.quare Bmotari fi ^tponlo cae e:ea eft b ao f. buco ergo ein bet fiat —gretfiaclnofacitatetbmennbibfacitgtergoBsctanircfitMomoaaogvtcabe.Tfiaiter&eiRb facit geteinffadrb:ergo^o2tlogabbTtbabf.feb;pbaraeft<ppo»iocaoevtbab ^ergoptopot —cioaabgvtgabb.cftis^rlntcraetbnfieruegmebtu8 4>porticnatis:qfieratbcmofh‘anbam G^it tcrti® inf CO0 mcDf*’i)poni6alf ccdocrttiDuo ijlibct n0cfi cft fapftcialcs ItTcs Gll^tecfic5uerfapcebfiti8.SinteniaetbficuncpbuonQeninterquo8fitcmebi<’4>portl6aUsmko ^ —aetbroBfiaalf8rC8n{e8.c8ptoenlbenilmo6in4>po2tionea3bc dqmimiperbeciminonitertif numerabfitaecccqualftfitcpibfcfimfetftfrn&erabfitcetbcqaaUdntcpibri^mgmanifeflGiraip clicilafubbuobusUrenbuebetf:etbfubbuobusegcotineaf:aetbBbiifinition£e(rcfnBfidaIes. fiebq> nntrtmile8p3.nl qtperbmotbefimbrtenumerltcetbrecunbSgtPnumeratercfimgtcte —noRieratcfecunbGfTtcrpoftrfiefl.fgirurperTictCimaquintirecunbibfetegfnntproponionales: i^bffimt latera a»iegfunt latera b.funttgifaetbperbtffinttloitfifbcficialesnftsiqfiengiporttfi. VI 41 CSifiicrttouonuainiBfia3lc3lirc0:cnijip<’!“''’>'>'’?®^ltt!ft3^5£>ialiaoqDi3tu CiSmcact bouo fuefuloles rtrrUl«6:oUo ^iportionc aao bdTe qquaoutf a& quat)iatii.nani Ciinte pmiffam crit tnter'to8 rnos mroius 4>pomonaU64ftt c. Sumoergo treemminw* tn eo? .ppomoe (JfmtfefupJtmi butueD-rferiitquat>:ari>ct cumq4>poztiooaoecaf»taa»c.tqcaofeafircaeb ergo B^jpoztionailtart 4^ipojoo oao fquaozati ao quao«tli eaeft aao b:q6 dlptopofiifi. 4i CSi ouo2um numcroium p2opo:iio fumt quequaDzari aD qiiaDWtumilloe fuper* ftcialcafimilcscncncccnc clt. Ctt^ccicftcoHcrfa picccocne.Smt aet bquaotati tftt flpoztio cao o4aa» b: cieo conumeroeeffc fuBliciales fiK».Warn bcozre.arw quarte bul” eavrme mcot«4>p«tioali9 tn/ai b.qrc tcouooeci nuquartivn’ erit mcof* ^pomdaUa inf cct o.qrcfi 4ol?**‘*c ®fi«fid4lc« riG^tqd ifipoMi 4jeSiogo? rupcrfiajliu llmiliu alter maltera oucattquaDtaais numeras pKioutef. CStntaet bfuBftciatca flmllcsmucaturcp aitib “t jentat coico cefTequaotatfi.flam btrfeeftma nona bmua inrer atbcll vnuo nicoius ^ozrioiuUe ^^cottnae,ppottl5ale8 per viccfimafcjtaiu fecuntu locni cfl qui ftt e? ambtey btn fe.at qb epo tn r« fit dt qaaowtfcigititr etequoe ficepamb:quoocRp:oporiTum. 44i dSi bT Ducru Duo2um nutnerojum vniuam alterum pxoDucanir quaozatuatfllo? ouoofiipcrftaalco fimilcacffc conuentet. (nft^ecconuerritpzccet)ente>Ttriccatnbp:ooucaturcqaabtaru0:bieoaetbefieruplktale8nmiU8 mam per vicerimarc^taiii Tccunoi latus tetragonicumc erit mcoius piopotrionalts tnKr aet b.erg» per quaotageftmJ 3ecb Tunc Tuperficiales riiiuIes:quoo intcncicur. 45 $r Si DUO numeri fupcrficulcovni fuennt fimileetcrunt cl inter fc fimiles* CSlntabt>uonitmcriruBficialcefimtleoc:oUoactbcfleimerfcnnitlc0.'8lain4aacrimllisctnC aerceRvnusmeoiusptoponidalisquirito.et 4abetUmfimUt0cinter btefimiUtmnus inctuns p:opo:rional(8 qui ricc.iaimr per Decim^fcctani q^rtt et inrer atberit vnus meoiusjipotttotuliff quareper quantagefimS puius aet berunt rnpernciales nmUes tet ptopofitum. 4«eSialiquis numerus mfupcrffdalctnDUcacun pjoucnicns crit foliDus. CCfiamtuperfictalisfircjoilcrunumeri In nomerfifacta wu^nultipllcsrione.qiii igitur ejt Puetu alicuius in ipium ptooucetur mbusUteribuo c4rinebacur edtinus ncMmalcipUcatione< quare per oiiftnirioncerit roUcus: quos intoiortMntr. 47 ^mnisfoliousabaliquorupcriiciabnumeratur. C'V^nt ots foUbueper btffinicione babettria latera quo? tertiam bnclF in otobncttf eje reUqois^at talis ^9uctuseftraperaalts>numcrafigimro{8roUou8^aUqaofucltcUu4tt»opond»f* 48 CDatis tnb'* nueris cii triplia mo foliD®jJcrccCivnus tit iD«p crit (J eje olb*4)DUfcf CSmrirta laceraabcietpziniorcafnbfiafbtTetcinbfiat Mimidi.MoefbineSatfeteftinf fiatfoliouog. tertio et aincftatb er epb in bfiatfollouslt.rico tres folibosegUe?tribu» Uicrlbr mpUeimocreatos eoroecfTcatc? cquiU8.'illq:altnb facito: etcinbfadtf> ergo cocrauaCecunbl oaofvraaDc.ergoBrlcerimaqiitni.ffccuolioccft^fircraifetcinotfcosinffacugTcinofaate Tuntimf cgrolioieqiiales.^ieq:ecbincfitfetccalncHtb:ergoBcanoloctau3que,ppoitiobao ^ ncadlfaob-crgoEvicefimlqmntlfccunoiiferiiicemfictef ainfetblnb-^fottalnffitfolieusg: er ej; bin 1? fit folious U.funt igif foUoi ctUaotuice cqualesrfeo et foliO* eporitrdlequalisfoliDo g» §2c6emrciaml2ercfuntfoUoicqlcs.furiiigiFvt 4:ponebaPtre8folioieg{tat>lnicfeqle8;i,ypofiia 49 COmnis numerus atribus numens numeratus foUous dTcpiobatur. 4£Smtabctrcs nficrinuincratesoiolcooelfcnuiiiepfoUofi.navclaeRpmusrdppoflriJlppoiit^ ergo Bquaoiagcfimafetta bumo 4cun(p nfierus in ipm oucaF^wucet foUoBtficcp cofubit ptopofitfi. g_ St a&t rit piim^ qt bnCerat ofm aliqiu numerfitergo Bquarti tertfl anumerat b:wl altcrfi fm quf nucrat.quare baut is fm queti Serat eil ruBficlali8.erlt ItatpB pt*n® ®foUe*. cOlmouo:? rolioo:u 4.ipo:uo:cc]c latcru vm^ aD latera altcn'’4>oucta pponioib®. U^Sint act bouo foUoiifintcp coelatera a;ct fgbUterab .oico ptopotrionf aao bcfTe copofitl et *“ MOttiouibuscaoftDaogtrcaob.oucocnlcinoet^enutUTTingefptoqeniatl.Enonaigitnr '” quinti ptopottio It ao lcopofita efl et pzopoitionibus cao ftoao g. ct qi per quaetagdlmaoctaua; buius fi coucatur in Uct bin lptouenift atb-ergo per nona qmntt ptopottio aaob rt Uao l«eao b*o ptopottio It aolmonftraracftcopontacr cao fetoaog.ergoptopoitioaaobcomporitaeftc); ® ptopoitionibus caofm ae g:cteaob:quoo eOptoporuum» 5iCSi OUO foliD» files alws duos equalit numeretnUos quotp foUoos fiTcs ec nece c. »- CSintabooofoUotfiinllc8tnumer€t«?occquaUtervtfecunofic.oicooeparttereffoUoo8fimiIes aingnoctiimfgbJatcramctblmlaterab.oucoipfingt^meniatn etliinU;poentato.etq:binn *- Bquaci;agcfim3octaaabaia»fadta:etminof3dtb.eititaopninafmb:ct8inofmc.ergoBvicefl» P- mStertia ptimi ncfl in ofccunofi pioouctfi er cin b401 p.ct fimiltter q: oeft inb fm met bin c fm c: npcreanDevicefimarcrnaoe(lmefmp«»ouctfietcinm4fii‘l*'tmcinbfacitpct<mmfacirq.ergo f" que^jpoJrtobaomBfrptimafecunoicaellpaoqtquaretqfaoUetgaol.fcopinnfarttoiqino facit cvtmonRratfieR^seo t>fieRfing4>tumn;etUinI^roo.ergopfginfcfadeto:etqUlinre “14 ~i4 -1+ b— —II I- -4 f-- —3S— —1b- f»dente/imteT5o6‘ter<sUoictUterftOftmtpfg‘rlatcr«cfum{<Ul.fcttilUp:ffbatafunt,ppo:tti> naliiatergo oatTotu eti3<tmile0»iuoo dltommpwportra monfh-sti}. C5ntcr quoflibct ouoefoliooe fiimlce iduo numen mcortnua p:opo:iionaUratc mcog mtcrciDunr. C0aincpenuUtn3a>P^nonlmta;tiomo'mia8Ulo?roUoo:SaOrel>quflun9>1arcnsipfiU9 a9 lanie alt^e triplima. Sonumergo qaamio: niicroe In^ipoirione loierie eo latus, tqi ^mleo qturtfi trit aiponio leteris eo lenis tripUrata: Ule ertt eeoS^oinonifolloi .to foUou. er^o per tmoOttiinaqntftictaotmeoiafunttnterpitmfitvltttnDm tot erunt tntcrr«UoutnctroUoum.et meoUftoMtoiergotntcrfoUoumtrollodittD^cmntouoiquoDcnpiopofirfi eSifuOTtit quamio: nuiHcri cormuc p:opo2ti6ak6:ouo cptrcint erant foliDi files, fj CbfcefIcSuerfa picc«>ltl8.Slnt ab c? (putmot numeri coiiniic^ipottionalcstWeo act ocfTefoU Oce flmUes.Snmo cni cffbquatruez lllius-ppoaionle nwwos (| Ocfi alTignatte Uoc fuerint cum o feOam bul’eTblbitcubtmyactoerfitcubi.quarcrolioiltmUesffc©firBiailabit#polItu.‘po:ron lloemtMllntrt et bPft quinti quarti nntcontrafepiimbquareper vtceftm3terra In fuajiponiDe mimi et B»t«n»3nw*5ei“n>€equaUtcr nnmcrabfit aet o. feo e tbcump fecunoi buiu^nt cubt ipi funtroliolITmllcs. ergo iterfi per qaiquasef{m2pam3boin»*etofuntfolioiflB»:qbcftp:opofrtu. CiSmttm ouoiu foUoo? fimlUutdlppoznovniiw ao altera tan^ cubi ao cubum. 54 inSbuaetoroUoiitKsmico^onionlaaPPelteptopomon^tan^cubiaocnba.qtcquinquage flmifeninoJboins Inter at omntouomeoO^ipotrionalesJintergo illibc: capio vtpimscfgbm Ula jpoirionc mtmoe quo? bfeSam bui* collat e t bcfTe cubos qll iiol fint cfi polrtie pftat ^jpoITtQ fTnoioteoeiqiceq^j^ttoaUcateq jipoiftoeaobcobi ao cubS eaoeeaaoozqditcrfiejipontii nvntusdD alteram fucntpiopottio tan$ cubi ax> cubum: quoflibct duos nu^ 55 itiCToa foliDos fiimlcs effe necdTc cft. ^l5«eftc6iierfa^cco#tisqlo«ewaefrccoutndf:q»ccon'eUrifiquartcbal«infquonibctcubc»8 ouo infciptatur meofl qipontonalcs.lgf couooecima quarrt et tn( illos foUoos ItioS ouo nifiaccbfir mctlj jtpom5ale8.ergoc 55 bul* ®uo extremi qerfit folioi afllgnatl erfit foUbiflltssqo cft ^ipofltii eSi fiicrint duo foltoi vnt foliDo llmilcsiipfi quocp mter fc cnmt iTmilcs. 55 ^CSlntabooo folioi flmtlcsfolioocmlco actbeftefolieos inter fefttnilco.TRaqt acfunt folioi inf fefTmilesxrgopcrquiquageftnufecunoibuiusintera eefuncouomcoq^ipomonalcsqfintoe.ct ITmillter per cano^ inter bete qnntfg. igitur BOectmafepttma quartiouo limtUfcruntinter at b. quarea tbper quinquageftmarertum buius inter fe fune folioi iimiles: q6 erat ocmoftranou. CSifuerint quatftio: numen pjopoztionales et ouotum cttrcmoium quilibet in 57 qu3D2atum alterius Ducaturtptouenienteubi aliotiim. esint ab coquattno:numericStinoe^poittonalesdlti^cquaOurSa et fquaOtatuooucaturcpD ine 14>ueniat get sin ftpueniat b.olco igitur get befie cubos b t c.ouco ent bin fe tflatli:qt etIt fnntquatitatiqnop latera funt aberit per tertia bui^4>ponioeaoUtan7j)ponioaaoboupUc9t.i: quare can$ b ao oiergo per vtceflmiquintam fecnnoi ioem eft qui fit ep bin ti et ein o.feo qiii fit cc b inUeftc!iba8b.quiaurtej:efnoeftg:t0iturgcftcubu8 b. fimilftcrnoucactir cinfet flat Icrft per eanoitcr^bulusppomo Uoftan$ aao bouplfcata.quare tan^ aao c:qu3reperrfcefim.5quint3 fecnnoi tt prtus ioem erit qui fit et cin l et ain ftqul ante fit et c in icft cubus c: tqui fit et .1 in f eft b f^turbril cubus c:qnoo eft pupofitum. latcnbua Iluc latOT jjpoJtfonfliiisbbUoo? ftimma biniome® mudhffarc. 5s ClbeclnroUotBottnulscqerOircretiscanolbSTeriatlscn-timotne.fiptimooctHrlateraoucquS cun<9ftlatnfeinuicf mafc5 vm^ettria alterius et ^HicmentftatimBquabtngenmloctauJbmiisfo lUMifi rfime.fi autelaterfippottioneso^rcopone per tmiaquinti laterujipomoncszfiuceas per fti^quinri continua et upoitio et la^rfi jipottionibus copofita per quinquageftmabuius aut qet cotinnatis cSponctur: em^po»io folioi ao folioS ftccpnora erit rolioop bsbictioo arqt rorum quoo 4>poftrf0puponitnotum. Scjnidementop arithmetices ^oioantfinis. fir numerus dl qui in duo cqualia DiuiDi potdt. ^mpsr eil in quo sliqus ppzinu pars dl abf^ paritaDDite^ rnp2aparem vmratS.t^an[um numerotu •alios pariter panalius panter ipanct alius impanter par. -g^aritcr par cft que nullus impar numcrat.-^aritcr ipar cft que quicuncp pares numerat :cum fcdm Iparc numerjf.llniparrf par cft que quioa par fmpare tquiDa fm impare numerat ifecDius inter duos numerus Diaiur: qui inter eos pofims equalif ab vtrocp oiftaf* .^CD^ Dicunir ouo numen qui no bab^cs alique mterpofiturcquales babent sd ce ircmos DI Ifcrcnas.'perfectus numerus appellatur qut c)t omnibus fuis partibus c6 fanetis perfietnfr.*? bunoans Diatunao quem omnes Tue partes c5paratc:mamo b.a Icrc rcpcrifitur.t?iminup’ vero cuius partes coiuncteoes minus ipfo conftituut. ,CSi quotlibct nbi coaccnicnturpares: qui pjoucnict erit par. dllameicojffHiinoeparisqiiotqiwtaff^abotureapciUlwtbsmeoiettrf.etodaeo^inrtfetaJea ere]ci3pumlaoinuic€comliae:coaanunoimi9ulcotifluufit, coaceriutuatgirur cum insuo eatis otuioipolTit erit par: quos uiienocbsmr. 1cSj paret imparcomungantuncompolituscritimpar. dlHamfiab impariauferatur vnirascfifoUmrmtat^pcrclfnnlrtffnSfuperpariasMKrclinStnf par.qui fi aosamrairmpartT)cr^esente eompofitus cru par.c«iqmolfi ablata rdhtiufwSaet totue per sifFUtitionc fiet impar.atillccrit4«paritiparicoftat.rar6eIl ergo qtfptopomtiwbau 5CBiimpar impan aooatur :pzoucmct numeruspar. CftamflabwocpcotumDemanjrwiiraerdmqucturouoparcsqui per SteptcmtlTamcofticBent numeraparemtcui quibecompoftcofiiungarursue ablate viucatca quepar£coniUmuntper eanse anteptemtfTam tetua rurfum cuaset par. boc aut^ mtenoebatptopofitio. 4(Bi mtparca numero parco fibi coaccruerur :totus ck cis coflatus ent par. CWcanfierua quo eoafumuit’ ftt par: in eo btnari* aliquotiee fumet 4toties cpcepentibinsrio copuUtterfit parca.tsif per pnma \}ui^ixnmici ftniul aooirf pftiiuunt numep part^d eftovofitfi. 5CSt impares numero impares coaccrucntur: copofitus ertt impar. dTHamftrnuseopcmittaturpcrpteceBcntcccimpofirueerltpBriergopcrfeampipfittsnpaoflto aooaturomilTus Ipantoiua 4moenafritur cj: oibus compoflma erit impar qS cftptopofiiii geSi3pari Dctrabamr par: reliquetur par. ClBani fi rcluiqrcnir tmpar :locm cum oetracto aooitua per fccunoS ptefentte cOTtoilTet impart: quoo cdcontra bfpotbefim. verum igitur cfl ptopofirum. 7<lSi impar apari tollaturirefiouus erit impar. CWamftrefiouus poneretur par :tocm oetracto appofitus per fecunOS bulue colKtutlTet tmparti quoo cd contrabfPOtt?cfim. erit Igitur verum ptopofitu. »esiab impari impar tollatur: remanebit par. CTHamfi remaneret impar: loem ablato aooltua per terilabntua coIHtullfet parem: 4t pefttuB cft impar.rcUnquitui' igitur par:quooenp:opofituni. 9eSi par impari fubftrabamr: reliquus ent impar. ClHamfi rrflouus elTet partfubdracto aooitua cpami bulus codituidet pare: quoo ed contra do» fitum.relinquitunginirptopofitunjvcrum. (O pacet iparcE we tooco erit nuppiuartparin ^ctoquoties vnltas in mMpUtSu^ voitas in eo ponifelTe in nnmcro paruergobqaaria l?otus lUi imparesmfiero paresfimuiaooiti oftimont ite» parcmiquooedtotiimpjopofmjm. CBumparin imparem Ducamr: quiptoDuccturcritimpar. Clinam multiplicatus imparper Oidtnuionttotiescrfr in ptooncto quoties vnitas in multipUcSte intpari.arinillovtiitascdtn numero impan. igiturperquintlbuiuslUtunparcsetiuimeroqul^ imparcBfimulaooiticodituentpwouctfiimparcmiquoo cftpwpofitum. IX G^ucmcuncp par numerat:cum clTc parem ncccHc Clftamparnomer3sc®iffututonepot inouo cqaaoiuioiincuiuemcoietatcs ftnumer^fecunoC quenumerat oucaf^jcr nona pami cunoc p«>ouctfi^Ducir.ar loc numer’ ui alterimeoletatfi ouci’ gjo.icti a)oncH oiniioifi.tgif ^joucms vc 4pot inouo equafitiri BoiiTmitione cft panqS cft4)pofitii. 1} dtSuicuncpimparemnumcratrimparcft fiponas q* 4eum numerat fit parqwr pieccocnf?numeratus eritpanquooeftcontra bri»® tbefim.qut tgimr^ numerabitfi quis numeret: erit Impanquooeft ptopofitum. ,CBiimpar imparem numcraifccimiMim imparem eum numerabit. ^CTft^mfUmparnumerarctimparcfccunofiparemrcrgodUcmarfpcroctauapamiparnumcraret eunoerccunoftimpar&quoDBDUooecimai?utu6efttponibile.vcrumtgimrpioporLr&. ,5 eSiimpar parem numacr. fecunou parem cum numerare conucnici. ClBamft impar fecunou impari cum numcrarct:crgo pvnoccimam bnius totus edet Impar.at po fituscftpar.acrioerctitatpcontrabTPotbefim.reltnqucturigiturpwporirum* eSi paremimpar numcrci:DimiDium quocp ipfius numerabit. C'Hamft imparparem numeretq>er pzemifTam fecuno&parem cunamerablt.cum itsqi omnis p4t meoietste b^tTeatimparergoilleperilUus meoietatemulriplicorus per octauamet nonani pitmi puoucet totius meoietat&munerabitigitur impar iIle:numcratioimtOLfi:quooefti>pofltum -fSi niienis Cpar ao alique numqj fticnt ptim^iioc quoqj tao aus ouplu ent ^m’ C^iftil^raritptimosaobDicoipffleirepumfiaocoupUib.ninnoftierirptimusaoccrgoa'*^^ c5icabunt.crir igitur cuit^nittoninumcros coiter numerosa et cmimerastqulfit e.a4aonfierat a'rcctceftparqioupIusb.«toeo4p5tl»»ttoeqttapartmttgirorBp2cceoenie oetianumerabico DimioiOc.quare aabcoUabuncquoo cft contra l?rPorWlro.mt i^apzimus ao cqgcft ojpofitu. c+ Clt9mnis numerusper parem multiplicatus: parem ptoouat CSiwfpofitomm ab vnitatc numcrotu fcornou naturalem rcn'cm:omnifi oupli fu^ i» mantunpjoucntctomned parceab vniMc:erunte^ panum codc o:olnc fumpte que et mmicronim.ab tmeate oifpofiiojfipwpoiDoncs. C]DifP(m»nlUlle^oeabvniMt«recDnDfinemrsi^^c^innTRiM8abcoef:qQo?pe^o:^inS^omoDu 'pIo«99trlmnt>.M£Qp{linoQMpare88bnutttevr(podreaccet’tO0*re(;fiooq{)2opo2tloRe8runtntt .mcft?aimi{iatcot(’ponto?:vtaaovmt«tSa>aoa;ca9b:o so ceaoo-tfaoeitalceefleperoiPinftn <RbioeptrwnirRb9V0ttbap)7tUoitetitaceten>p.lP>2imapatet.namp}imooupIo8e(reparee inantMfid?ciimqutUS«teo2fltnint>uoO{in{Diaqueeau<('untotutoipofrir.£t^nu)ru8aI/usab ac S'Ci80cparcmiieqqttaUe«dl7«bercttneo(fiiniuimer(9:a’qbaberctvUaco(ndocretdiaUauo L0itatop<4cn6 eft ergo ois parea fuifle acccpto6.Secunofi patct.iia que4>po2rio aaP 1?: ea dtvni rati«aO0.er0ocmuratipermture<:fiPi44>po2tioa8Dvnitate:eaenb9P0>?requcbaPU:caena SPt?^0o eeanoeqbaoa:ea ett Uac b-et ita peanu^conni> OKcDe itotS fimupclu9eefipoflt&. CSlnumero? ab wutafc naturali fmcDifpofitom: ouo toiio conriiiuc coiunga tur fiem ois fpares abvmtatc.ipforucp impanu paoyimi quiepterunt contra fc p:imi. C^tponaC otPo numerop abvnitate^ conun rtatural£<er(cm»olungamr(p qulcb t>uo ipuo pu> primo omnesab wiitatcfientparc0.fc6owroquof<pauos4)xunoe elTccontrafcpJimoe. patet.nl qufoinqt Puoonrlml (n naturali ferie nomerop p9ifinitionerola vnitatePiUatTqre eop alter e(l par et aUaeipanUid ioirar ilmul aooltt pfecunoa butus tota conOttuat impare.Seo q> nollasfuiinparpieterfUosftccotlatalioqutnnDpofrctiilcimpar bmitiin pmas partes quapvna foper altera aPOicvniratlquoDeacontra PilfmitioRg.aut fi PluiDatcoflituetur et puobus numeris in naeuraUrerienumeroi^ptopmi8qaenptopontum.SecnnoCp3tet.nacunio{s impar ab vnitatc aoPitavnuate tvnitatcfubtlractababcat drcfiportto6paree:et ots par cirefipofitos impares ab eo foUvnttatepirilrespercoceptionetgiturPtlaquorGaincp^mopimpartueflbinarius.at binari^ eit numerus pzlniua cd anuUo alio nGeropteter vnitate numeref. assitus isftur binarius 4eit nu« memspiimusvm^coilituetpJimfitmparenumeBP«mutn: 4<lti«Pembin3riusnumeni8tmmu8 aopitue ternario numero CTiapumoipcrpnmatcrtj} coftituetnumepasalrcnimpjtmQ.^tiPeepcr noni terti ai^netabcrcTita oequibu/ltbct Duobus^dmia ofienoere pfaalecfi;q6 eUpropofltu. nyi9 m—'—14? 1Si eSinumero? ab vnitatc ^poirtonaliu fccunouo abvnuarefucntpar reliquos oes pares dTcig» fi locm fuerit impar: reliquos inocm Iparcs effe ncccfic cft. (nmnrecScuseapar 4anumeraf ab vnitatc fcSusnCcrabittertifii tertius quartuiUctpsctceps quarep vicrflmlterril primi fcdus (j cllpar nficrabit oes reliquos. fun ttgiE rcUi} oesp miooecima bui*pare8qPeftjjpoiItfiquoat>b«*ctnfc6uscftipar4anueraf ab vnitate: io iprelnfcOuct<’nfic rabit terrifiet per vtceilmaquintajfctfitertius ftneuno^qutreut ita ©ctnccps.lsitur per vrioccima buius eo2umquilibet erit impar:quoP eil totumpupofltu. manter quoflibct ouos numeros quo?vtcrcg par cft aut vterep fmpartvnus cft me « oms 9fi vnus fticnt mcoms vtrucp par£ aut impare efie ncccfic cft. dVia fi Tterffi furit panicaITmul assiti ppttmibuius collituunt numerS parcm.t it vterip fuerit impare tertia butu8is€euemer.n itatpilUusrumafmesietas pfeSam partlfecunseprimi illa erit nficrusaquotlU numerie4omabatBSifflnitionf.l0ifnumer^iUe erit inter illosmcsius.io^m p; Ili It li^ 04Ocmesios pptimi sarti feSe ptimi isc swUcare(t tanq> illi piacti 4di fit par eo cft sapl«naltereopntpanpre)»3pnuisetrdtqat}s.etRlaltereoplltiparprcprfm3eiurs£Trdtquus quoseft p»>poatutn. Canter parem et imparem vcl nullus cft mcoius vcl ouo: q» fi ouo fiiermt mter cos “ meouialrcmm parem talterum imparem cficcoumcitur. eSivnitate silnterft par at<;ptpar.inter cos naUusln/doctmestu8.q>'ltlpUus<{tuncpoiftiterint vtparaettparbisuoinfaebeesuosmesios.na precunsabuiusaetbl&ctipftitufitnumepiparl vt C4{per nonaetufSIsiuisi poterit in ouosvnitatesiftate84 ftut 8e:quos stcop tertii primi ^ui« Btftareaetb.quareperSilfinirtoniruiumes4q4eftpumS.Sd>osico 9fIlnteraf btuusuomerl alterfieireparlt^tepfmparl.nliisuovt9einteriammes<i[*.Bsiifinidonlvn{tateoiftabfit.quarc eop alter eritpar et alter iinpar.coinncrtlsifscprcSambuiusconftitueru impari, atabeoniuneti per tertia ptimi coftimanteanse.qnarefiaIterfuerirparper nonam buiusreUquus erit impar.etft alter fuerit imparper oeualbuiusreUquuseritpanquos eft propofitu. C^nicr quclibet impM^e et vmtatem vnuscft momsuoci? cft maio: ipfius pozrio. ^fi inter vmtare zipmvnus mcot** epitcririquclibct tale nume? impare efie pueniet Sit aquilibet impar et bvnitasr suoInter atbelTe vnum mesifi qui eft mato: portio a. sico etia fftlnter a7bvnus (it mesiue numerfi aefte impare.’l^rima paret.nl cum anumerustpar Tupercr bviut^ tn numero parivt (n c4eft sUferltta aas b.et qr ccft par oiuisal in sqo equa sMcfe.capio ergo numerfi fcuius cflt sifferltia asb: que sico circ mesifi inter atb. nl 43 per conceptione siffe^ rentia aas bcoftituta eft ec sifferitlis aas fet fas br? siiferetta aas bfit c. erit ergo ccoftitutus ee eDiferitia fas beius simisio et saltero cius slmlsiorquares eqoalis esiiferetta cft af.eft igil Vn fnic?iusiiitcratb.etc((3etrbatvn(Utfrupcrct'UFtrentijeiuBabvtritateti:t0f (ImlUtervnitstj . rupcrc;cHc4ConfhCttruBCi;pamon(btt9eF.Brf<^dlmet>iu8nummiBmau»cftpiKtio‘ZeiniROi^ ' conactcaqi inter citipareTvnttat£dTeTn&met>iu:eunOeq$dTe varisntaioi^p«:tii>n£<)tfenptoil. Sc6miTRtcr.twninteratvniratcmbfltfme6lus:pcroHflnUloeniaat»f«l>a8f«iiuk9ft«itofie feo di per pcepdoem tiffaitiz aat> bppofTta JTt c% oifferfr0©aat* fet b a» f.ilia cfi ui ouoeqtie oliu di pofrtticrffpar^tlllam numcru0avnitatefuaat.a Igitur per blfftnirioncmeftimpar.qieiifc6o| et totum pmpofintm» •4 Canter parem tvnttatcmTcl nuUue erit mcocusJVdouo mino:(5 paris oimiwiura. ^IT mter ipmtvmratern ouo fuennt m«5t|Jcum dfc parem necefie dt. CESitaqu{{lbetparfTbvnlca0:liafltpar;p;clmereQue0Vfutatemc6rolaTOltate9ltott:tnteraet vnltatem null^eTltmeoius.9n3pllu8vnttatet)i(leDr.blco^moinfaTbbuoee<me»io8(intnoteq; eife paris DtmitiiS.ScSo^fliteraQu^aiiKpnSerumfbvnltamnouometMipflltertncae^parem, ‘intimum ol^iturnam aparieiabvnlmebtUftritia eli tmpar.^ isco bparem timparentmitatt dilates per nonam ItuIus partiri potetUportlatur ita^ilta oflain ouas tales pottioiies.tfic capi» c numerumtailusquioeminotpottioftt diifRetlaab b^jerpceptigemmaiotpottioerfttHlferdua 30datmaiozpo2tiominotevnuateruc3t.capioergoRtunerumptinuere<iBenteinc(tntt>.»UferAis eni aat> d:mtnot erit vniutebiferiaab c.quare biifer^aao oet cabbffit equilcs. Aint igttur c-romeb4 interaetb.ct quiaperfcSamptlmicencncbtetasb-rbPiecabbilmaliiiaoiRetbf bfiac abbequlturazcfib^tbfiacabbabbantvniutanaaocctcabb^igiturcminotmebibpmebietas parisa^eibbceiiptimum.Sc^moectaratur.namnintcraaTnitatembiilntbaomcbllTtcbiqaopc fit mino:fetomaiot:briacaobm(RotdivnitatebFiaipiuseaba.espigirvn3eflpar/eC3t/a impar» abbitcigttur fimnlpcrfcbaml?ulue pftim6tnumerum iniparemiqui per pceptioneroeftoiffereniia aab b:cui iTq&e aobioerts vnitatemzpfKtues a.enisitur apar.quob eufcbm» >5 CSiimpares oes ab vnitate Difponantur itonocmpoli quflibct fumatur :quotus ipfc ab vratate fiimt: cptrcmu ab lilo numerari pu«nict.et itan totus ab illo quoms ipfca ptimo loca ^roo nucrabitunffcip odnccpo.oefip in mcoio relicti eruntptiini. C.S(nc3/b/c/o/€if/g/if/k/l/m/it/o/p/q/r/0(mp»reeabmitatepfajaeteroifpofitt:etqaac<piotvl terius vble0»btco ptimoq> 1! ac^to a4temari^ efl ptimo abvnitate imparetquocus ipe eab vnitate totum imparempofi ipm acdptasTtDribeIltcrtiummi^numerare.et<Ttotum iteppoiibaccipUs »t g:itepai^ninuincrare.et11 totum poftgi>rl2iteva.i^inun>erare.tioprcqaenter>iSc8onne9ios oesvt bcie f/!7>tTetitacSreqn2teraoaeffepzimos.t[brtmnmp$mam4at<mariae£clu8elli^vnt> tatettree poftaccepti unpares et IpTetfont quattaot.quarebirer aternarium tbtertiam poHipTum funt bet c buo impares mebg^ttres bifFer^eeqoalesUfflpariumetd ptimeozvtaatopaoimpares: bifFer^eft bfnarius:brie rc5totibcm quot tunc vnltatcs in Uk>»reb rota bFia bab apct peeptionem bi6repctitam:pponi£eictlUsmbu0bilreret$s.i^tura{nvnamillap buctus.‘4muctttotam.atipfe toti biffererie abbiiusioiUtuit b.igtmr anumeratb.ctfillrer elKblf anumerareg-font enim qutnqj mfmcbQ Impares tre):^binardbFieequales:bupte4betnbu8pam{0.etDftag ab aSPpoiTtaexilto fcjr.ergo aperoiccflmamtertiam ptimi illS numerabit-atatoti bftcg ab feabbirusoiHtuitg: atgtS numcratg.ct Ilptcr nficrabit I.43 brial ab aerit tripla bficb ab a.o(mrer<:Bnfierabit o»4a btfferraa quabrupl3.ct l?fic in mobum pfcquctcr.Scbm pj.nam pmo anonniierabtt b:qftquib« Detracto aoe b.per buobcdmamptimianficrarctbinarium biifer^amrcnDuamnioiff5muk«cm.qd eftintpofn Itbile.nccpBlbem n0erabitc.namipofubllraaobcnuerarctolfferentiamc3b3.qnccfilTtbtn3rius bip fumptustlT rubllr3J?arur sb illa bria numerabit rciloufiorlam rcil5 efie qeritvniwstmaiot fal5 mino2cmjetnufnmiST>nit3tcm.qb eft impo(ribile.5t‘iQi'o‘9.pbablsc 7 ft reliquos intermcbjosrf ptiiTios ab amamff anfierarct ctfubftrabo cab g.et 4* anumerat gvt olidum i/teptr l?rpoti?cnni numerum fubfiractumjergo per buobedmamptimi numeratbfiameabg:qnc€pftjtutaej:ouobtts binarii8.qtfnupcrmoftrammeftimpolTibile.SlBterfinuincrarctfAjbllr3bofabnnmcrog:ctrefl5 btiuB ent binarias vt notum elht anumerat grbctracmm fier^o nfierat refiouum qui eft binarias maio: ininoterti.q6 eft impsRlbilc.ft ita ^ba bn&erarcg/n/ s:fcb intermeotoso^s efTe ab b{Imos, eicnucrarelctstinftncoios abipmcfrcp:imoe.l?abcbif<?ofimilivbiq5m’gumctitarioe4:ponnHn. attp ita l?abef impares relictos 4poft aUque roto? toti nonvcniuntioft btrtbecimSl^m’ elie ^nios. eSiab vniiatc impares coaccrucmriqm pucnict ent quaoratus. CSitvmtasjifuootomcpfcquAes impares a/bic/b/e/f.bicovnitatemrapftitucrequabramm. ^cmvnitatoiia tbpftimere quaoratumiet vnitatcm abet c.ct ita pfcqueter quotquot ftf aggrega ucris.bittiboentaptlmumlmparemlnbuaspotrionesvniutebiftitebtetcertueftmaiotembiftare vnitate ab viHf3tc7mJot«nvnifattc4cqa3lem:quasvijoo:binelttuoiu]crap:imamvnif8tem certfi eft buos elTe ctoines ab vnitate fumptos vfcp abbuoevnitate biftates (imttatem eni bic nomtenftert cenfcmu8)crgopermcenmamtcriiamrewlvnttastbu«tUepottioe8aggregatepfhtufttquaora^. at bue ille poniones equitur imparivnitatcmrequ^i.vnitas Igitur Taimparrequcnsftmulluucta: :oftiruutquaoratum.1itemrefolutoainvnitatemtbinarium:cnmblnariU8fltbrl3bab3:brcfoUie£ pernonambuiusin Duaspouionesiquapvnaeritbinarius/TaUabinariovnitatematoiiibcftequa a-q cfiptima vnitateftaruatnr vno otbinc iupta i>ottiocs8.tflccrfit buo ojbmes abvnitatcab noeros vfcg witare biftates bifpofttitergo per cabem tcnlam ttrlcdlmam fcjti iilt ftmul aggregatiplUtufit <- -3$ “53 -51 -t7 -iS 4S 11 3tf 915 7iS 9 54 1 mn4 0tf Stft m— 0« •••• e•••e ab••9c 099 •e abf •• «c CSitaqpiffiijimmrtrue <iul ta altum boftua bjjcucat cnumerum cfemtjn «ut abunbaicmji b nuFncrrr9.nicoQ>ai>acnistno^ucftnficrumab6oanrrm.n2mqu(acfeptim9m recfioiq^ipoino baaoeafftcatftfrt bpcrbrpfliljfiimnficMrcUfilmrtcnirmcrabUf.cafitpofituadlBftctusaut aboflpana.tffnur pqninciua^lTnlqDmtBin bniue feabunvabit.quot» c(l ptopofitum CCumbifpofm fucnnt ab vnitaic numpantcr parce qui piucti fcccnnt numcrwn «o pumumjqtrcmuem lUum ouctuo ^joucct numerum perfecrum. 4L3Uivnitaarr8bc 99ttnue bupUtqui fi viceftminonam bmue erunt pariter parcett coaceruarus CTcmctvBifBteftteqDiponafeflepttmuaettcttrnnua partum fuca? mcounaggregatum tjiue niatf.&icofe^QunKrum fifcctum.manifeltQmcftmtpemceflniSrcmampmia/bcuouploeriiptoe «6numerare Wrqjit>fcUme;gbkvtfc5onunicrctffrfme:cfc«mgibfctfinb:erafcbmh.quottc« ergovnitaa mt) teticg ctn het quotice vnitae in ctotic® gin f.quarc taoiuuate vucftmarccfioa frfl tequa ^tpomoalitatequotiea amo toties gin f:et fi ioem quoties btn ototies bmf.cr quortee cin oiotusUinf.fclncfotcptcBruntc/gbkapfi^tctnitasabcaoo.f^ctrlceftmarebambui^uas ctabetcfmml mm‘'fuittvnuatc5'ftto.quc4oc vnitas cftfisooenoUtaperconccprtocm abo.crgo egbkfininlmimis faciiitfrnaetusfitcDcnoiataanunieroo^arsaiitfoenoiataa numero oteil e. funt Igitur eg bbbmul minus ftn numero r.at vnitas abcet efunt tan^cigitur cg bbvniras ab c0Hinul tm (unt ficut f.ft afi tfplurcs partee non babeatteondat fi oiffinici^n ipni fnumerum efle perfectum f(09 nullam alum babeaioedaratur.nanirtbocritpofl'ib& numeretergommumerum ffcdmn.menicril^mus qumficratbntimcrabitcrgo per quartam tcni|m aut n. quare fi vtcefimi quintam ledioi reliquus eo? numerabit o.ct cfi afccCous ab vnttate Tit nUmerus ^mustergo cvice funafetram quarti quldi(?nlieratoeda{i48 ino 20inepteceDenctumn)teba>atrifircnucrat rfciim g;et(ibrc<inb/etftarcbmbpartC8fe5ptcpofitB8ctTtOicmnitll.ccn(lat ergo nullum alium a poi ctisnGcrareF^uarcpoicdruntoesruepanes.ciipfirimuimodrattruntiFmconiplereatcpfiltcere. eonlUtergooilfintrioncm numeri perfecti et pgruerctifrnjcTillereperfKtum.quoo ed4]poritumf CSeptimteltmcnieiumSrHbmcncc8 5ojoani finis. 02mc numcjojum c; Tnitatu oifpormoc cpcn&uttir.C! Cft a0t wfpofirio fp:ima in longum tmtaltcra mlongum tlatum: rema Ho mlogum/latumi Talrum.CflCums vmtarum bifpofino in longum rifi cft: lincang numeruo mcimr.C^n bmm oifponurur fupfia.ilce imeanbuo vnoiqj pcluITtquo? vnitates Dircctc 7ao fer ao aliaa cqiialco babciit Dinatuiao.£U5 fit ouplicitcr fc5 vL’ vt vmtateo vnifatibuo oirccte poaprcturivcl cap infuallis opponatur:itavr fingtor ojornum tcrmmj mlarcnbuo rcliquto «rapiatur. c5” ^l^um ^0 Difpofitio foliooifi c0: coaptano mfimui fupfiaaUbuo:cir quo:p latcrtbuooirccrc fibi appofitto folioom fupficicg crcctc conftituutur.Ciflumcrotum mrupcrficic Difporitoaumrquionm funt mangularcd tna babenteo latcra:al0 quabnlatcriialfi pctagoni:al0 plurtu laterum. C^uaonUicroau autqmoam latera oppofita babet eqiUDidatia: quioam mmime. emumeromm autem quo^ latera oppofita funt cquiDimna: ali) oia latera babent cquaba ztetragoni oicutur.alq fiquioc latera babentYmtarc Dtflanaa 7panealtera longiotee vodtunout ampUuo7oicutur anlogiotcg.cflumcro:^ maltu ppofito:^ oiticrfag cfle bafeo eotmgmquotfioa tnangulao/atcci quaoragulao: ct ita dc cerena. (Lf^omm oim quioam babent latera fimul applicata: 7oicfinir pframioca.quioam babcnttatcra:qucctl!tnrimultcnofir:non tnpuenifibct curte pframioeaappcUanf. qmoam babent latcm cquioiflana 7cqualircr CTccia:quifupio:cm zmferiotem fup^ fiacm babent fimilco.^ftom quicucpbabcmbafim tTiangulamtferratilceTocanf: 4 quaorangulam tipft mlongum/latum/7 alnimequaliter otflcountur: cubi oicutur. quicum^ io bafcababucnnt pentagonaa betragonao fiuc plurium laterum:? latera cquioiftaniia creoatcolunc nominentur. C^ionumerud triangulari$:ma babet latera cqua!ia.'Slfi manifefluih cIl qu^ibet f triangularem cpnumene ab vnitate otome cotinuo oifpofitto:clfc conipofitumratc^ latus manguli effc fend numerum . |t3>'<tnbcvnitate0baltstnanguli:aba>?rc:eracertremirares.Oicoergoq’inrecfito otole cuius enTcmttare&funroetcnintfiauciores vnitares.nemfIrotiOcnifurnntin otointbusactoctlateraa otccen3nteqoi{Vatta.qiftvnitates tn vtroqjoiomeequalif otflanciicc^non cocluocncthaAgalum. pauciozcs ergo erunt tn otoinc oe.cr qifi ^rum oidat aab bttfi oiilat aab o7cab ctfi ergo ooninir f totaabcqnocaoabctoidabitfabo$-tumcabc.ettwoqnmaabo.fi ergofnonfuerit 4>idmapon qfcofuerit bvnitas meoiated otUent aab o^rum oab b:ertt longituoo inter aaf^ta inter a7oet tnterffovramplutsiricfpvnuaresctaorcinuicemetae alias non cqualesferuabQtoUHttas.quoo edtmpoflib^vtoocrgoocminotel>otoincacvnitatc.et eaocm rone ffi otoofequcns minotvno Vtii Q-pccccoeopoKCircrtrciiiCfu vtiUaa.ctqtioco torirem erunt e:oinc0 quottnbafcfantvnftatfe:p5 retroagenoo vnitatc cfie piimfi potentia trtaitgulti cui ouc aemnete fccuncucUuPuntfnangulumi CUI fecunoo treo aoouc facient trrttfiiqi femper dl vnitatc niatot ptcccocti. et ita cofequenter Wm naturalem niimcrcifi ptogrelTum ficip ptcpofitio tcoirelariu nota funr. 1e:©i aquolibet manpbri numero btuo fuum oemb-itur; relinquetur iriunsuluq ipfi ncfcoiDcnoo pjoeimuo. CSttagiilufi mSgulooefccnpcPoptorimuoicftcuiusUtualarcrc alterius cftvnitatcminua.atrtS gulo ergo abfubflraao latere abiouo qreimqutf triagulus ipft maguto abocfccfcnpo p:otimuc n.t q: triagulus 3per Repente babet ma latera equaliaict fcqufria fola vmtate vmeenna.fubilracto ergo latcrcabcrfit reliqua latera vno£6tractio:a.f5i,qtvim9tfi oitlnesfmvnitatSfemBoecrcfcuf P3 per co.TcIariHptcccoctiorcf^iouuclTcmagularc.Tqm latus ciusvnttatcvinciaUtcrcfriaguUab cognitu cflipaiet relictfi triagularem elTc ptorimfi ocfccpcnoo triSgulo ab:qb cflptopoftifi. 5«riOmmotrunsulartebuplicama: pane altCTalongiotero effiat. eSumaturaet bpuo rrugiilarcscqualesiapiidafcp lateri etrianguli a b eittrema vnitae trlJguU b.itib^ lateri ftriaguli baoiiciaf ecjcrrema vnitas rrtigull a: fieiiri^buo latera vniratequotlibet la^ tusa tnagiilop 2et bfuperatia.ctfi acois aptafm boc pacto mcoios otbinesvniue mcplis oainib’ altcrtu6:^cn tper tertiam ptitiii eopofiti otoitics libi mutuo et ertremis otomibue efet e0 cqualcs. verbi ea vt ft oiomc feqoentc latus eaoois abaptaftp oioint fequenti niltate 0.qi enl per coirelartfi ptimcbuius 0200 fcquens latus etanta oiftat ab eq>tum ojoofcqucno vnitatc© Piflat ab ©.ergo per tvTtia pJimi lllt fimul luncO equitur oioini e0. et ita ©e aliis mcoiie 02©inib5’ fi qui ne©0 rdicti funt tUMccrgoeoplctuscritquaoiJguIUB eoius latera oppofitaeruntequaUatobbocecui©jlUtta.vtqt latus e fcquaieciTc nioflrata cft latcric ©et mc©iie:crgo vnitatee lateris b©cquioitab '.t vnitatib*» lateris ac-t fiiniliter q: latus b©cquatur lateri eamlfunt Utera triangulos equaliini et meatis o: ©inibusilatiis efcqutoiflabit lateri e ©. 7qt Utera b©et e3oppofita tequisifiatia fmularera trian gulcru airign3to?2T©uoreliquaefTc©rmuUtcr oppofitaa equioifiatia quotum qiwtHbetreliquo alio:a vnitatc funt maiota. patctpoiffinifioneroiangurSab© eecouobusequalibiistriagulanb^ cofluurant''quare te]ctri3guUriviio©upUcato)efre altera patclongio:t:quo©efiptopontfi. 4CCuiuflibct oUcra parte Iongio2i0 1mcDiemo cH m3ngulari0. (ClbcccflcouerfapmePctis.Sit altera partelongio:quicunq;abetvnt]mciusloi*<^DsUrueab< ©ico eiusmc9icratceflerrUnguUre.©eUtcreenim longioiia b©etrabovnitat£aet>eereeun9oo2> ©tnc ©ua8:©c 0tertio trcsioec quarto quattuor<7 fl plures funt tot eofequenter auftra quotus oioo ille fuerit abvnitatciper coireUrlfi piimcbuius torus ©maenis vt aeftfl rrianguttris.et per i^nt q: continue minus vno relmqudiafvfcp a© mttatcmrrelietus efi triSsularis. feoet qi ©etracti Ums cfequarurrelloutUfcrlgb.namUmsgcfupcrabatUrusgbvnamttatctfcOTgefupcratUoisf e vnavnicace.cquantur igitur gbct fe:quare treliqua vnius reliquis alterius cum per pntnatpuiiis omnis trilguUris fit cquiUf 1 totus trigonus aeftoti trigono abgequalis-cfi igitur ©atusaltcra partelongio: in ©uas mcoictatcs ©iuifus quap quelibct efl triaguUris: quo© efipiopofurll. 5Ci^mlibcr DUO triaiiguLirco p:oi:uni:tctragonum conftiruunt |rSitrtfnp:cce©eri9bgcquicun{ttaltcra>partclongiojqufpcrean©cm©iuip3fur inbgbeteaf ©t'O0 trilgulos equales. crit vt acfit latus trilguli e3fct Ume baaltera ct© iongte<:is 7fufi oppo fituicg maius Utere ac7fequctlbus oiPinib^^nitateBmmoto ergoUtere aeremanebit bb equale latcribltfiiS«gb.7t©coquaruo: ilU latera 9©inuice©ifianria:quo© ergo per ©iffinirionerelictum efbcfitcrragonus.fco 7ammoto Utere aeab trigono acfper fccunba boius rdmquiFtriangularis truguUriaefp2o©imu8:quarc7trianguUrlbbg libi cquaUpiojrimns. confiat ergo tetragcniieb 1: gb e© Puobu8p:o©tmip tnaguUribussquoo efipjopofirfi. <ClP>otcftqBP<5ct iPdTenomm quo© mccftmoquintocapimlo fueSritbmettccspiuus affert SeuerinusomnetetragonS fi ei p:opiium latus aoiiciatur aut ab eo ©ccrabamr alreraiparrelongioiem fieri. TRatneumtetragonus omnis vns ©iquac^Utcra cqualiapoiTlPcatt fi cuipiavnumiatus aoiiciatur efficientur Ureravnitatooifiantia potrofiUms viiuma quopiam ©etrabamnrclinqticnirUtcravnitatcPiftlrUiquarevtrobiip facti eruntaltera parte longio:es. iSiuo fit iteruvt pjorimf qtticp altera parte logtotee amcoio tetragono cquali oifferentia oificnt.lBam cquali Utere fubfiracto 7abotto. jCHatCTcjpoiTto numeri maguloris fummJ muemre. 'Slnocmamfellu c(l Hoamq lltnuracnie imporfurama triangularis pmiicnirc fipans ptorime maiotis fubou. pium mjpfum impare oucatur. et II numerus Iit par: fumma triangularis proucnict fi paris mcoictas in imparem oucatur ptepime maiotau. CISii a©atu Utus cuiufllbcr numeri trilguUriscuius fumma querifiouco a(n b ptorimu numerS maio:e 7per ©iffinitionc pioueniet altera parte lot^iot cuius b maius Utus erit vnitatemaius 9mi ©I b • f* • e••0 co o o Ooc o a bo o o 000 000 Og ac •eaf cr•e ©•• bo oo 000 bo ooc/T a••tt ef •r• •elr ce ocoooo noKUtcre tUro-e triSgnU cnludrummalpetif. pnquar^3 ergo bums rummaFrliguU petiti eft me* stetsesUerapartclon^otieinDcnninctpccnnst^pofltuni. IElCo2rd9ri&e^^emoih‘atio^econcltt ftonie BOiunlte Dccimapiimi notu €fl.a l^:cc6 cozrdario fuo piawm ^relTlonia ocmoftrat. c^mneperfectum cll mrium. C^cmarttts totus eft religione facer aa“Sirrtjagotic; ZPagIset Saccrootib’ ollm in arcanis mis rumtnmooumocner3tu8.rtSrtftottle6'|piYtbag<Mico¥fymboli8lnfiften8co!poica magnituolni ipiuniB Tntuerfum perfcctfi aftrutt «j ipfum trinfi ftt imoTcro trinfl cftiqz iprum perfcctii qtJ pjfo Tfbyrtjagoias copcrit ui numerisinumeretfii* mrfterlle.ljic ergo fufcipif Dcmonftraoum omnenu* merum perfcctii efle trigonunt.Wam per triccfimJfccunoam fcprtmt quotcuncp nfleripariter pares conumerata vniratecoaccniart eificiunt numrrfi ptopimo fequenrepariter pare: rnitate minotc qui qnioem piorimerequcne panter par per vicefiminonamfeptimi eft coaceruatojfi maei/ni ouplue: quarcretuscoacmjaruseftjmpar.atpcrfcFagefimafcpttnuoienumcruepcrfecruogignimrcrco acrruanoneTnitarie et panter parifi quoties coaccruareft ptimus qui vt oictfi eft pTicefimanonai reptitnifola vntT9tcoiftatap:otimo panter paruqui ao aggregatoptnatinift duplus cft.ergoma):i< mus aggregatoiu fubouplue paris numeru aggregato? impareptojciniefcqucne: in ipfum impar€ pucnisrpcr ptlma partem cotrclarfl ptcceoctie coftirutt numerfi tnangularc.eft igitur ofs numerus perfecTus rrtagularis etep trigonusiqd eft ptcpofitu. £teius latus ppteceoctem tcotrelariu pitme pujueerutllc impareFomnibuocoaccruatus. •••••• ••••• •••• l_l_| 5i 1 1 uT;4-i 8i m‘8 1“ — -J. 0 ' »0 b** ^ 51b — ^ I5 Ci0nim0nunicra0 parmcoiafiji parte tnulnplicatuo tanto mmo:cfumm5pouat «omneofummequem iprofunicoaccruarci^tumfuaiptiuemcoiciag. CI^Ketanalrtteisnumerotft Pcp2optaeft.cteierc0aparrecoireUr4rupio2ioeocIuiIoni8 abiuulte nonapiimlt coi oignttatcnmitatemquccunepnumerueu«3:ipfum ^«bucercftari cognita eflepot. eSifumantur triangulare© ab vnitate: ouos pnmoo impare© ct fcquete© duo© pa rc0 aicg Ita viddlm DUO0 ct DUO0 unparco atep pare© muentn ncccfle eft» Cifiula enlpcr coirdanfi pume bui* tri3gulare8crc5tur epniieris oiolne cotinuo ab vnitate bifpo fitie et vnitaspmue eft potdiarnagularis'! poteria impar, aooitalgtfnumcrofcqu^qeft par eo ftiniet Bcoa^darlfi piimebuius fm triangulare tper fccunoa fcptimi imparc.crunt ergo ouo piinii maguli nGeris boc pacto oifpoftrie fpares.Sco tqt tertius numer^ eft ipar ct cu p:ereoctt triagulo Bioemcon^claTificoftituitmangulfltpceocno moftratuscft iparicrgOBferriafeptimUcrtiUBtTia» gularis cricpar.etqtquartusab vnitareniieruscftpar cj cum pceoeti magulo parip tocm coftituic quartum mangulum.crgo quartus magularis per p:im.1 feptlmi eft par.crunt ergo ouo fcquentea parcs-ct iifocm rationibus moftrai tur ouo Tcquctcs clTc impares iviciiTim alij ouofequetcs pares nct^ftneoifficultarecertum apparebit piopofitum. eSi poft ynitatem fumantur numeri mangularco :inter duo© ct duo© Diiiuncnm: erunt que inter numero©ab vnitate naturali fene: fumpte pzopottionc©. CSit^mustrlagularispoftvnifatgatbcocfotDineprcquetesfiitpvnitascrgblinCcrtabvnt titenaruralifcrie4)ce»liee:oicoqucjipo:ttogaovnitatceIcircbaoa:etqDaoge3drcoaoc:Tquc b30f7C3eflcfabe.-tim4>ceocooinbtsoiltl!ctim:tniUisTerocoiunai.namBp2ecet>ercmacfttpar bcparc8;oetparc8:ctfpar.fumocntquattoojparc8abvnltatcqfintgUlmctmn>uintcreosfms pares rinrbno.ct qtgii eft binariveft latusaqcfttem8ri9.cpercoirclarmpumci?ui>fj latus bt It latnecctniafusotlUfuseetolatusftergoBquartabuiasginbfacitalrcrafircIongioicsupiga «binltwiPlftb^tUtnouplficctnlnloupiao-tUnoUHplfieiati^pcreanoeoinrtiouplfif.qtcrgo gin bct Iz in bcoftimont ouplfi ao aet ao b-erttergo per octaua fccunoi vt que ^potrio Ii ao gea iit oupUbaotinplfiarfcOabaoafaboupIueaoftibouplilpcrcoirelariuccdmcqiiarrcfecCoitlcuteo» ouploifi oupluaao ®uplom.ergo ilcut Uao gita bao a.ct per loc ficut lao Uita oao c. iilcutmat> I* ilafao^‘rcfigU^m^lmfulnpti ab vnitate cotinueparesicrgoBOedinaoctaua fcptimi eftjjpoitto baoavtgabvnitetiTpabcvtbaosetfaoevtltaobWu^^i^cftpiopolTrum. esi tres continui triansularcoftunianninquirubcttreintaeonancnireritoltera panelongioa :cuius laterum maius erit illotum trium rneoius. ©<eSinenb etrea continui triangulares tcozfi latera ocfifircp gvno maioz f.oico qfl fit er ain ceiTc r+^ ~alrcraBtelongio^cttffflaiuaUtuaeritb.naqiabcruntctinui tnlgularcaiergoEcozrdariupzie c5b'-' ®buJos^effuwfmcomiHuafcrlenumeropfumpti.ctqzperquartabuiueoirtccoftituittiuplfioba c>*5a’gofiocftparqucabmoofibiccft:meeieuaoinecoftiniita.ctpcrit'cmmcciicta«fiiiccoftiruitb. ermcoictaafmgcoftituitc.ftaStodrctimparnumcrarctafmmcoietateci-rf bfmmcoictatce.ct fic cofequenter mmcoietati pariumIparea numerarSt tri3gulare8.ergo per ortaua piimi aoiuuarc coirdarioocdmequartefcciloiqueaipoiriofaooteaeftbaoa.ctqgaoeieaeftcaob.^qzabc fuiit conrinnefumpritnagularestergoBcozrdarili pzimebuiuebfuperata^tumeftc.fumoergobvno mino«b:mamfeftucftbfi*Pci^cc*9tu[ncfto.ctquo abfioatcfuperbcftf.tquiabaoafieutfaoo odambaoa.etficutoetracruaaooetractuiltarffibuusaorcrtoufi.eriiergovrficutbaoaifacaob totnaaorotfitquob iglf fitccaincBVicefimaquintafediotrftieftq>tum qbfiterbinb.atmanifcfti! eft q© fit erbin bdTeaUera p3rtelongioie.igitur etqi5 fit erainc.cft altera Btelongioi cuius maiita Uterti dlbillozum trunguIoiQ mebius.ct piopofitum. IO Vin par Ct impar ;Jcopoflto «ipHe m(no! aftoaturl itcmcp ct bis confecto mino: aoiiaamr:cnt vt cum pzimojij ptimus (TCparrTcquaca rmnmparco.tcmojii pumus paret fcquentes impares ac ficoancepsi wtploffi oua D:an panum et quaonto cjtrcmi impanunctragom maguUres tcrmmftur. CSiiit aet bnumeri primi poft vnttaripar timparbinariud fc^ atai temariiw 4ionetl fac&ccal a9oUoaiuto.itcmcetocdponentecuiabDafc-tristtan9etfcomporuat0eteaot>atDrfft&ub, Pico ptimoaelTeparc et bimpare teo imparcautcmcparlfctgl? impar», acnteflcparemibim parem ert brpoti? ajccopoiiirur esat biperfccunoifcptimicertl dTimpar, et «ftoconflat er apariete impari per eanoerccunoIocdlmpar.TqreroniUrescoouobua impai^ua eper quattl «ufoc erit par.etqzfconftater CTClmpart aeqi pari perfecunoaetafbifeft impar «per eeno^erfit ect bimparceiqd inteitocbaf. Scc&oo oico ouplu quaotari aterminare tctragonicfi rrianguUrem. terminare cni tctragonicfi triangulari ett ciTe latua quaozati qui pariter cftt triagularie. ipiatet io nl qzvnitas oupUcatacofhtuita etabie Tunipnie Tuum periicttquaozatfitouabueigiC vnitatibuaa eirtatafuoquaotato etvnicumfufdpitmcomfcjb:cutuBOifferiti3ao9fft»nitas.etqtperfeptim5 ferti que sbunoatquaotama bfuper quaotatu acfl quoo ftt ervntraie in bet afmuL feo bet «fimut runtvnomaiuaquaotacoaquioupiueeracao a.tglrurquaoutuebcflvnomaiotouploqaao:9tia. feo rqj ouplii quaoiatia In quaotatfi bcum fola vnitatc otftcnt ^ocftper oiffinittonf aitera parte longiotc quaotatus aqui eil mcotetae in quaotarfi bper octauam fceunoi j»ouat lUiue altera parte logiotis mco(etate.ergo per quarta buius quaourfi ainquao^ti bptooucit triiguUrem.Tq: pzo^ oucitur er ouctu quaozati in quaozatiiiergo per ocrimaquartam ferti ioem erit ttctragon!cuo.cffe; etum cfl ergo qi5 oiccbaf.Xcrtio oieoquaozaruoertremi imparifitcrminarcfetragomcfi triangula rem.rumo mi quaozatii equi fit k-t quaotatii oqui fit Uoico IzoupUcarfi erceoere vno l.nam quaoza tum couplicatiz per oecimaquintam pzimi eit tan$ oupluquaoun aet ouplii quaoza» b. et quoo fit erbinaquatcrctqzpcreanococcimaquintam quaozarfioeftq’tumcinrcctamfc taincbistfeoc infcefttitumpercaoeinq-tumbfnfcctainfcceainbbia.etaincbiaeft tifipernonlpzimi^tum ainfebisctalnbbis.crgo q6 fit et oin feeftg-mm quoo fit crbinfcctquaozuplfiquaozanat qua ozuplum ^ouertcr ain b.feo cum ouplii quaotati efit mondrarfi^m ouplum quaotatl bet oupIS quaotart aet qDaozBplfi quoo fit crainbtcterfecunoa parte buiuaquaozatfiberceoatvno ouplum quaozati a.ntergovt ouplum quaozaticcdtiitettsquaozariib loco buiueq-quaozatSopHnetouplii quaozatiaccterw omnibuacdfimilibuaerceoarvnoquaozatiio. ouplii ergo quaozati cmquaozatii 0peroiffinirlone pzooucit altera parte longiozemcuiua longiue latua erccoitvno quaouttfi o.iUiiis ergo oupU oimioiue qui efl quaozarua hin Iquaozatfi opzooucic illiue altera partelengiozia oimi? oium per octauam fecunoi qui per quartam buiuaeil triangularis cteuiue latua cill quaozatua cp ftemi imparium qui ct ioem per oeamiquarram ferti eil retragonua. et bac ratione oc fcqucnnbus molire io quoo ptopofitii cfloHenocre: et omnino conilabit ptopotitum. P» o oo o• o o o o o o 0 o •b •••o0o •••oo o CSi abvflintc continue DdToncntur parte altera logio:cs et (iuaD2ati:DUo xduo eo n iuncnwangularco omnee emoent. «TE^dlsnoabwutafcabcqnawaroectabwittatc alma pamlogioTCfl qui nntocfitJicoquoflibet »uo8 ijnmofl qaaotatfi et altera parte l^ioam fiitiuUcolhmcre truguUrcni.na fi fuerit p:int*qu» Biiiuadtftvnitaaaprtmue altera parte lostoi qutcftblnanuamamfcnCcftiUooflmuUooitoo b coircUrtIi pame Initus cdfttnicrc migular?.» «fuertt afetius quaDiaio 15pumus abwitate3lfa Pceld«otf»peTweftm5fmlfeptimiqtut>iatu8apftare>:lparibu0abvmtatefc3ctvnttatctpno ibamutft ^iwparMuarc ft rcfoluaC aerfit accepti cStiitue nunicri ab vnitate. bcoirclanO igitur ^bm*pfctufit m3BuUr«.t iraoilcM aet e cofttmere triplare refoluto aiii futs iparibus bitf fcptiiTUtftnfui8p8nbusBWcam3fmaeiofi>«.ctttai>cqmbumbetDuebu6bocniofumptis. CSifumantur manguUrco ci tetragoni: fticriniqj lUi toti triangularmm ab vnitatc r. quon ifh tctragoiio:i5 poft vnitarcmieonmncnfaacnt numeros pentagonos. CHamnwiltanquccft pjimus tngonus aptungae piimfi quauiatfi pollenitate qui cfl quafnan* ftet petagonuB poft vnirate habes quimp Utera equaUa.ci fi capis fcdni trigonii ab vnitatcbabcbit binanfi mlatere tfeefious tetragonus poft tntta te habettcmariu mlatere, abiecto ergolait re qua uian fiet ptr coir^nfi piunc bul* ma^luefeques habens Utus tnum vnitatii et figura pe'tagona recipiens torioe in quolibet eius latere wiiraics. er qt femper torus tetragonus poli vnitate vno pUs In latere^ tngonus totus ab vnitate: bacroneoe quibofiibct alus ^Jpofitum efficere, tqt lattrailloiu pentagono?equaUsfuntUtenbustCTragonozaoatowmibincetia fit vt illi pentagoni tonfint poli vnUatiquorittetragonuC^t «hac ‘iVKefiinafeytafcpnmtcognofclEqdoiuv Scuc ritius alfcritbortias:i4captfcrccunotruetnftitunois arithmeticcipctagono? generatione nafci ct numensnaruraUferte omnitis quSuocdtinue buoatnonaoeouopfempcrmmeinofacra iiucm.if fione lungfitur. 'Bam per gentem accepta vnitate queeft piiiuus trigonus tquaternario tctragoii’ ouoB in mcoio binarii fcf atep temarfl facta infiniffione tunctisifit fcefious petagonus ouiMU latere gcres.Btrurfum wiobus tnrmnilTisreptetunu8a<ccprusqutnanatncluDittbinarUjm.Stqiiina nuswnpaTperwceftmJfcwamfepttmiaooitus tetragonoquatcmanotpioeimficoftumtretragonS tbinarius vniratiaoiecms per co:rcUnfi piime piotunuvniiatt coftituu tngonum. ergo bpiefcntd »mt8S4n7Coftiruuntpentagonn.'Bt rurfueeuobus oimtfTie poflfcptcnariS accepto oenano qui yincUibirfeqiientciinparf tiemariutniquifcpffnariiis quao:atop:ioiispenragoniaO0!iuSB vice umSfertamfeptirotfucceirfttmconftituuquaoiatuetternarmsmgonopentagotiiaoicctuspicpiii quum mgonum.erit igitur per Diefente numerus e? miiaic 47et to cenftimtus pciuagonusti ira be quibuflibecalus argnmcntabcre. CSi paitc alfcra longto? cum toto poH vmiatcm triangulari conmngamr: compos u finio cnr pcnwgonus poft vniiatcm totus. CSttaaltquows ab vnitate altera parte longio: vt feciious: cuius ibco minus latus erunt rue vr.l tates; fitcp bmagulustotus poft vnitati cuiv latus toeo erit ternarius t equalc maiezi lateri altera parte longtozis. aoiungoergo maius latus mgoni bmaioit lateri altera parte longiotis a: ct fiet rc> tragonusfeotilluoquobr^atoempromaioteUtere bpcrfecunoanihuntB cft triangulus minot befeenoenoo ergo per ptccesentcm Ulecmn quapiato ep altera parte longtoie tlogioie Urere triana gnUcoltotourtt pcmagonosiquos eft piopofitunu C^mlibapentagonojQ poft vmtate cum toto triangulari ab vnitate: bejagonum r+ poftvmutcmftimUlocopofitu conftituit:ctbuiufmoDi tctragonicusDicamr. nunieri potcftatt optfnerebtcif cuius equalia latera bellgnatur. hincipfa cft piimns triangulus ptimus tretragonuspentagonu8b«agonus titabetnccps.et pieoictotu numcroiutn ptlmus poft ipfam eft cuius latus fiimt binariusitfccfious poft ipfam cuius larus fuerit ternarius f^busvero ab vmta te coios larus fuerit binarius.et bintur hetagonus tetragon tcus cuius larus eft equalc lateri meoq tctrsgonUoico ergo q» fi capiatur aliqnotus pftagonus poft vnitate vc atqui verbi nttfapoft vmtaifftt tertius er btorus triSgulus abvnitate: bico 3cum bcoftitucrc betagonil pofhmitat? torum pariterep ttetragonicfi.nam per ouooccima huius quotuspetagonus fuerit poft vn itari et toto poft 'nutari tetragono conftat et rnlgulari ab vniiarc toto, ergo cqualis trlagularis qui eft quS piopd pwponit ct oppofito tetragoni latere appofitus hetagonfi ronftittiet: cuius latus mt equalc iat^mebd tetragoni, et q: latus eius fequalelatrnmebi) tetragoni cftt cqualclarcri pentagoni quare tD«agonu8toros 0*0 poftvnirate quotus er petagonus.tqi cius larus eft equale Uteri meta terragom dt ibi ttctragonicustqfi cft piopoftrC.fic hac ct buobcctma hm’ 7coirelario pmt cognofee piomptfi eft gnationehetagono?eo pacto 4socet £ocn^ i5capite IcPi arithmetices C£^i0 tetragonus cu pjcccDctc altera ptc!ongio:c: bcjagonu iciragonicu pftimii. i? C1^zimuemim altera parte longioJpoft mirarem fft qui habetvnitarem minuslarus^fenjnbus quibj ODifsrier Ira beinceps-TpierebeetcrragonGfftniiusmmuslatuseftvnomimielatercretra goniqoi tpoii vnirate torus cquotus talrera parte legio: ab 'miiatc.bico ergo qulcuncp tetragenii vt aefi puceb^c altera parte longiou vt bcoftuuere hetagonuG mragonluii6quartShui^ altera ,VIII i^v.. uiv<u iriagDionim cquaieiicminc loniacltabvnimc quotus rctrasonneapoftvniut^viius ergo Ilio? mgOToX^smo^mSbai* tC icfra^no ai^ftituit pentagono pofhnnta te tototfeo et per pteccocntf^cm cfi altero tndgulartqnieftrotuaabvnttatcpfhtiiitl?cragonatctragcntca.aiguurcuinbaltw^artel5fflo*e pjcceoctceolhtuitf^cragonfitCTragoiucuiquoocrat oem6rtranODtnig>pofttuin. ^* ttf CSi Oirponanf mangularco ab viiuateitcmus atep vno ititcrmifTo apnus fumoio ternue: cnr betagonue. ClBjpcrvnoeciiiiabutuooifpofltisabvnirarcptcaltcraWgfottbusef quaOjarisRouoiouopimt gamr fient oes rrugulares.ar eu pumuo trugularis poft vnttatc fiat cj: vnttareT ^mo altera dte loit gioieifcflus trlagulans poft vnitarc qcft terttus ab vnlrate fiet cf {Jnto tetragono poft wutate xpnio poli vnltatc altera pte logiote per eanoc Tiioccima qui cR tetragonfi pteceoes. ergo op«ecoent^ter tma tnaguUriBabrnttatcel?eta0on’.reotcaogr6neqtvnotnfmtirotmlu8feniBaawaioetpce» qSocSpzopofitufT?'’'^ ^vnoccinia I?uiu8 nafdf:ergo per peeoente lerdu» femper crul?<wgona« 17 CSifcp vmcateo vm circuponanturtiiavnllam aoinuice tao illam caoim oillliia bciagcnum cquiangulu cortftirucnr. CI]H iqueUbet 8UC lllap onltatacim» mcoia mirate rriigulutn equilaterfi confHmunt.ficilt Orgo fej; triaguU cquilarert quoe oleo ena cfTc aoinuteeeqaiaguloe.na eum tllo^vnitatea abinmcfrajftoou oiltltpnitarczlimtrarcsvniuaTnitaitbfcuiulltberaltcriusappUccnf eruntl^uptafeinai^ appoflte et aguli miU8 Iguloe fllterlus no eteeoetes: quare fet agult IjewguU erunt cquales. ent igitur talts bcjasonuscqutdgulus introiiur.ct ita oc.tctragono cquiigulo tpetaaono ft ploccraofteoeref . i2 CE,arcrc p:opofito betagonum cquiangulu conftitucrc. dSitamttaree oart lateris qbus vna plures applico: ita fentg aug^ Oonec tot fint otOlnes quot mirares funt in Uterc.fiatcp Ijoc mo ocfccocuoo oonec tot eoplcanf oioincs quot vniratee funt in co oem latere, tqt fiunt tria latera equalia aferofoo et tria flnntia oefecoenoo p; creatu efTcbcicagonfi que Dico equiagulum.nlmeolusoiolnQroniBperviccflfnltertiSfepriniiiifliparqitotusritanGero laterie quotus nuerus Ureris abmitare.babet igif iUeniaeimusnutiterusmum meotn.f filTgnas eetrcmitatesfcrUterumperbcocfgeritilUmeoiamitastneoia Dn>ini8fctb:etototni8Vg.qi eqgalea funt et eacoicant.oiftat ergo eqnaWabfrcetabetbetabOTg. quare fcjmagulirfljwe DtamitB8fe:meoiamitasco;meotamifaoocaneoiam!taffeb:ctineo«vi»ifasbg:rnieOUwiitaB gr:aoiutcge4UteriruntetvtintkeoGriequiagDU.2tgtfbcragan*beeefgeqotagul*:4tlG|>polltii C.Si aiiUbct bepagono cquiangulo tomefcnarQ muUipleje aooatur quotud^c dt flb vmrate: iTmikni bcicagonii pioucnirc ncctlTc cft. CI0t{)mo bcragono^dlmitas fenarius cirefiponaf per penulrtni3fiafcqu£s betagonusequion gulus.na babes ouo in Urere q> fi eo^t interualis que ena funt fejcfec aptfturmttattein caoe oidan tUtarip ipfis fer viiica hb^ alie fce anguUrif: fiet fequ^ bibens remarifi in latere, xvt pzcccocs boragonus fecuous erat ab vnitate ira aacpras eil totus fenart multlpler:fcil5 ouplus ao fenanu qui eRouooenarjus: cui firurfum eirefiponas tripISfenaHI :fc$ oaooecf mitates inouooeciInterualUs et fec ao fer aguUres mttates cRiets fequdem tjerason6 equtigulum quivt tn ptcceolteptobareF b^bet quatemariu fuG Urus.et qifenip ttenialU crdc&t in fenarto tot S^lis fcmBmancnbusappofltofequctifcnarfi mrtiplicifemccfficiesbcpanguieequiiguUijeaoemrSne fequctemtquoD intenottptopoUtis. ao o o •••«.' bo oo o o •'••••• co oooooo e•oo•0 o• o •o oo••oo f•••9 •• • c oo••oo 0o•o•ob •0o oooo o••• d 6•o o •o 0•0^0 •'o o•oo•o o•••o oooo !o cesi offlinaitur ab vnitate triangularce ct Unguli fccunoii fenarium multiplicgtur t ptoDucti oopita vnitate erant betagoni cquianguli . mimvnitate que pumus triagulans eflfenario multipUcessfecfious brragonus nafcitaru|ui aooita mitatccirciielufaeognofcif rcquiigulus.et qi perptcceoettm quilibet beragonustoraooit fenarios fuper pceoccem quotus ipfe pceoco fuerit^ mitate. f(jli^ tri^gnUris pcotreiarifi pme tot b? mltatcs fupza pceoctemquotus ipfe fuerit abmitate.ergo flfetis triaguUns ct loeo b; ooas vnitatesftipta piimu mulriplicef pfer aooet tot fenarios fupta pccoetcm betagonG qui fuit fcdus: quotus ipfe fetf us bejeagon^ fiiit ab vnitate. ergo gpceoctem erit fequee bc^ffon*. xita quocp ftio triaguUri 4aboit tres mjtatcs fupiafeb; pi^e]cmIhplicato:r quolibet alioiaooes er niuUipUcadoc tof fenarios quotus rfriaguUris tptcceoes J^jagonusfuerit ab mitate; fitep ppteccoefem femg coneluoes piopoRtum. 1, CfiSmnis bcpgonue cquiangulus tanro abunoat fuper bcjagotmtu tctragonicutn fimilipoftvnitatclocofumpfuicptuacfttetragonusabvttitattfimilitcrDictus is CSttcrfpU ciaquamis'ab vnir8tebejrason’tetragdnicv4t tertius fpoRvnitatedittpbUtueab af vnitatcrctragonusioelitertius.-tquarrostetragonueettneolusalterapartelogtoJo.otcorcTuS ,ff u bcFagonfiequiangulupoRmitaretantCoillarcabaStuseflb terttus abvnttateterrigonue.nam co per occimlqulnta buius eet ocoponunra: feo xq: oilfcr2tU bao oper vicciTm5quarram fcitieil ter pg narius latus b:t oifferettacao ocft quaternari’ Unis c.erfit ergo pfeeunoS pilmi btcfimul maius b——c vno $ouplumo, capio etorum ab miwtc triangularem quotos cft btetragono?; qmaminua Ums bj 0 c e oo0o 0oooo oo o o o ooo ••• •••• ••#••b ••• •••• tUert pittelonfftoTtso eH Utue b: 4uotK)uiol locm eft Unte (.ergo per qusrtd buiueo erU Ptipiufi c:feo-rbcm6fhiT8funtwiomaiBa©uplop4'untergobcvnoin8iu6 qoruplo e:crg{)bcofimulfmit mo m3taefercnpIoe>8tm6ceeent2erei:lesrumpm8 aoeitavnttarcrerttupoflvnitatd creat bnai gonumcquiisulam quifit (.confUt ergo fer bcctotrebsTotu confiat c]ccDvtoflenrumci>.abunoat ergofbcpagonneequunguUia ababetagonoterragomeo toro poli vnttatco'tutii cfibroruaabvTiU tate terragonoatqS cft pwpofltfi. CBtumaoucrte qufaomoounv otuus moftrat feucrlnue beprago noe coflttnaefim CK brragonia er mfcrioiibua trigonis ouobus equis tferioribus eminus aDifrerie trigoiUs.qaob rleem euenit fi etTpontis natnrabi numeris: ettremi cdtinne quarruo: infmcotis re lictis tungatur.vt fecfibus beptagon* fit rr fenarie fetioberagono i eno: pmo trigono etn in*’ equali abificto mgono^ui te^ fir nmoet fenario quamiot in mebio prermiiTie numerts: ttertius bepta< gonns fit er Interno bexagono-t ternario fcbomgonocregiSeaoiiicto equali trtgonoberagoni qui lef fit ermo fenario i vneenario. et bocpacto in figuratione rcollectione fuotfi numerotfi in ceteris efiptocebenoum. r©upnoatum nummim ocrogonum cquilaterum conftitucrc. »* CSitaoarus nuerus fupia quf labemur octogonQ cquiUtep coftltuere.^jcebod apponfco otbtcs vno femper fccrccoctes bonec totioe flnt quot funr vnirates {latereterfircp facta malatera cqualia (tmariffloomuuabbo otbicsequaiesquouftprotfmt quot funtvnitares inlarere erfiicb facta qufcp Utera cquaiia.ocinbeab vinmo otoinc ^ceocnbo appeno oiomes c^tinueonltarc minotcs boncc tot flnt quot mbeto latere vnltates ficntqjficoefccbctibo maUtera foictls cqualia cruntcpCimul octo equaliaUtcraartpocrogonuecquiUreruscSfiimtuetqdcratfactenofi. CIBUaapubbiufifcuerMifi octogonopmucnifooram ratio mobufqj eft vt cotinueej: toro beptagono et fnb fc trigono furgaitr.t tbi nunt fi et btfpoflris naturallf numeris corinue in mebio quincp rdictisterrremi iun^rur.rt fi 7 Wobeptagono tungas wum ptlmfi mgonfi furgit oaonarius et fetfus octogonus.qut tb^ fir etvno ctfeptfquinq; numeris binario temarto quaternario quinariorenarioftcrfhtibusioimifTis. etfiist terno beptagonotungas temarifitfecunofi mgonfitfiet ii tertius octogettus.qut ibem colligitur ct 1,7/ijit b«pacto in reliquis cfi pwccbenbum CSi fuilibct bcpanplo cquiangulo poft vnitatcrtoiuc tftragotiuo ab vtiJtarc cum u frqultc altera parte logiorciungamr.pucnirtoctogon^^pofbftiitatcfin locofumpr. C^QtiiUbetbetasonusequilgulus CUIUS larus fit aet matimueotbmfi eius fit bqtiiabbat fup anumerfi c.qifi enimberagonus oaciio baber mlguloB in mcbia onitarc coicanttum latera cqualia: btncotoimsbmacimiilineaniinoiefivnoibuplo larerisacrgocmlnot vno^a.abbo ergo ab bo:bis ncscqualcefccunOD c.erqtabaaobfiuntrrtaUtcra rquahaficnrrurfueabbrDm abbirisotcinib^ buo alia cqualia quibus aobo relictos beiagoni pofl b:ftcntni rurfus tna alta reliquis cqualia crifi? fo:marusocrogonus.etq:prrnonap:imiclnbfm ozoines abbitosefitantumq-rumcinfeitn a.at qtf in fefaat efi tetragonue:et quob mafacit eft altera parte Iogio2.eum cet afola mitate bificnriet tetragonusiile totus fit ab onirate quotus octogonusfotmamspofionirarem.nambeicagonuo vno maius latus babet latere tetragoni:-! alteraparte longiozfequfs tetrsgonfi.cdfiatergoq^beragulo cqmJgulo aifignatopoft vnitati: abPltus totus ab vnttare tetragonus cfi fcqode altera pte ISgfoie colntuu oaogonfi qui cfi equaUaberagono latera beantotus erit pofivnttat? octogonue:t4?pof(tij C^oanarfi fer cquilatera? qualibet fmvmtatfi oifponem figuraliter eofbiucrc. C^ienim triZgulostttragonosp^gonosbeFagoflosrerragonicosberSguloseqmagulos! octo gonosrecunbfivnitatSbifpfiRemfiguralttercoihluereoefiberaaiccpUmabniue trigonos figurare bifce.ey quita tetragonosiet ouobeama piragonosiet becimaquarta hetagoiios tctragonicosicrec becimafeprima tbectmaoctaua bei:agonos cquiangulosiecctTicerunarecunbaoctogonos.ficib per.- uium babebis ptopofitnm. Claterc oato figurarum ptealTignataru cuitiflibrt fumma reperire» CSitlatusbarfiaquiponaFciTcparf mcbierasar!tb:ftt<scmino!vnoatbcobfionomatc:.abbi ^ *^c»®bnatc.-t ittrficabbiro abefiatf.etfiEreabbiro abffiargioucaFqibmbetrequcnres-t^iuc "«"tbUlm.qient ain 0plhruit altera pte longioj? cuius trifigularis lateris aefimeoietas.crgo p quartabninsbmebmasain bjmuctttnSgularf lateris atfici^boRcbitrummfi rrilgularislaterls Dati.etqicDrimulc5fiicuSteqalprecunb3ptinueftouplu83.ergobaefunc ceiinue^ottionalcs qocrgoRt^binepei^ceftmirecraferunrtcfiquao!arfia.etcfii9 fltli ofienbetUfummStctngoni Utensbatia.TqtqSRtntbutfpcrnonSpzjmirmrft^nimqfinrbinetin c.eteitmcinacofiituat altcraparte longiotF cuius meoietas eft magifinscper quarti buius-crgobmebierasainc^ibuctt triangulare CUIUS latos efte4*co et bin ejiooeittetragonfi 3qui torus eft poftonitarcqitctuetrfan^ gulanecabvnitatf.ergoper nuobeciml buius bin f^udttorum poftvnitarcpcnragonfi.at ille eft ^tagonueactptobucruseftUoftenoettgiturlfummampcntagoniUtcrfsa.-rqtqueefiterblng eitt^mqTOm quoofiteiibincetlnf.zoictumcftepbincfifri rrtigularcm totumabonirarequo: tus eirpoft vnitate pentagontm etb infficrus.rrgo per fecimiquarti buius bin geonftituer bera» goimm tnragonicfi torum poftonitate quotus -r poftvniratFcftpentagoiiusatbecagonusUIccrit pat^on STjiDuctusmmonftratitaipinfiimmembecagoniailarerisafngnati.Duco itoiuin fe-r >^nicftiatnqaierittorus abmiratetctragonusquotus betagonuatetragonlcus ni eftpofiviiitat?. aoboergo nietnfimul-zptouenlatoquiperTicefim3p2tmameritbexagonu8equiangulu9a.nr;m totus poftonitate quotus beragonus m.onco Itemcln aiet^Jueniatp qui erit altera parte longio: Vm reiiufio tetragonum crc5 n. acoltto ergo (ferum n t ptetragono fc5et Tequete altera parte long(o:c ^ l7crJgulocqu(3gitloocpifefimafert(45utug4)ucnirixiogoii9toruapo(lrattatequ(Jtuepoftvni«t2 /4 cratl?etagulHt eqm3gulu80.quareortogonuop:oiientrequtfjtqcr(tctacgon*UtcrteOBt( aatqj | u fummJpcutioftcnocs.Seoeflontrfuaalatuooatnnumcniefparctbccircupofmpareebiwnort fi1ij* 9 crna(ctetmc»ictaol)fitocrniceicta8cfitc:pcrfectinoapjimiP?efiniulc6(lit«ent8.aooopfuper ^^ aetftatf:truperfctfiatg:ioucoalticT4)Ut'niat>?qmcar3noncquafupcrui9irnttn53uUn8a.rt ^ pucoatiifcrpjoucniatltqutcriteiuetetragomioctamfr^JucniatbquicarancncquapnueCTit *’^^ pcntagotmsactoucoamgrptoucniatmquienipcr cccinuiquarfaiti(>uiU6eriibcragonu8tctrar gonicuelatcrieMiia:fcoetl?etragonlicqui3gulumcto<togonfiaqucrcvfpauloantcmonftratiieft. “•^ •6 C^)mni9 pyzamiscui’^ bafto m5giilacft:quaituci mjffulanbus ct cqualib^^pnncf , tres atribue latcnb*’ bafie triaguUres crtgCtur 4flmul applicatur arc^ coumiunt tbafiecfl quarruscriagularie.i;cqtnDuoctouo laruebabenccomunc.‘eruntomace(aterrecqu(Urcr(: quare ct cqualee.quoD igitur aiTcritur peroiffiiutionc ratum eft. 17 CSi trugularce ab vmtate in altum fibl continucaoapfcntur: ptoucriicl omnesab vnitatc pyramiDcs: tcuiudibct bafis crit tnagu!b:um fuoiu majnmus. CISmtabcPcFaot vfcp vmrate fc5 quoihbet fiimpti trilgularee cotinue vno minotee 4fibiinulc? co ojDincapteturapphccnircpuravtbfuBponaf aicrcfuBponaf b.TtoipficrcmfitucifiMPcipfic Pico pyramtcce abvnicatc ciTc fumptae oeo 7cuiufilbct bafim tuos tTiagulo? efie macimu.cum etii numeri tUmaltu conlhmti invnitatf conueniat per bilfmittone conUimut pyramiocterftcp pco*re Unupnmcbuiueccsab vnUarcrumptttriagularcs.riitacpvnit3te conrioeraeeacccftetpnmapo tentia pyramte.fi^ximfl trilguUrc cuiue larue cft binari» cutpfi fiiptapoTira ccaptataqi mirateap pofiti appareat equales timuli^ couciiutce triagularceierit coftituta fega pyramie cui^latus eru te cunpue tnagularte.et cu fint in altQ continue accepti tnigularce eritr tertia pfraniis cofhtuta cm’ latue erit tcrmie iriagularisict quarta culus quartus trlagulans erit latus atep ita cemccpe.reo q> cuiiinibet bafis fit fuco triangiilanfi mapimus boc io eUg» pptecepcntebafie cmagulans cuilibet mattgulariu laterumcqualis. fi itacp latus fitquintustrtSgDluopcotrelariCpunicbumscrotbua pteccpcnbua trtaguhs pfhtutus ita cr bafis quintus erit triagulus te% oibus infcrio;ib*c5ftitutu8 qui crir illius ppramiPismaidmu»;a ita beqoacfitp altera fu® coUarepotrft ptopofiru is dSi oiilibcr numero cqualiu laterum oes ob vmtate fibi rimileo in altu apponatur: qui coponetur ent prtamis bafcmtpiprius caocm clTccouaiift .fivero citm vnitatc bumfiiiooi ciufDC genens fomiom maltum fuppofitio crccuoep ocfiaai:quot ab vm MiciUms genens rcUctcfucnnnioncs curta prjamisnarcctur. |£lOccDontretragonice bafie 'pentagone/beragone/b^ragone/octogonc/ennagone-pecagotte: et quotquot angulosmlucria cobc ingenio quo 7ptecec^ oe rriSguUribua p^rainloes fonnaremam fibocpacto tetragoni ab alf quo pato op mirate vfi^ruppcriri aptentunoia latera (que EconrUni pmcbuiusenTrtnffgularcsnumertabmitatcconnaefunipttrmtlinvnliwrticcconcBn^Mtatppr ramiPe qfluiieitt cui'b3fis ent patus tetragou^fup cuius latera funr erecta latera pyramiPio inmu cacumen conucnictia tino oes tetragcnicc pyraiiiiPcs infra lUaclTcnt formate.7 ita fi petagoni beta goni et fcqucresbuncmmoPii aptetur petagonas beragonas ocrogonaa magonasireliquat^cpo^s formabis pyramioae Setia pars Uan neca cdnafiPcturquotufciic^tnagulanstt fenus abmitare cutfupponSturmo atcPictopoiPinequitusquart'' tertius ecrctiu8.at pcritvnitas gruis latera lUi'* fiiuul tenPatnon tn conumifirupco figura illa ppilFinitionfcritcurta pyramis*®fiqutrusquamis 7tomus foli ruprapoftrl maltu fuiflcntiochiilTct® ouo fetius cmtitas bcc bifcurta ptramis Pterref cpftoniilTifuiCrcnt tres Piccreturtrtcurta.cnta quotquot omittCTctur toti es cuna Piccref. etitapc curt'S pyramiPibus que nquapratie pentagonis bciagonisafcqueiibus figuris nafeCfurfenncnpO efl.cll ita® notum quoP proponitur. ip CSioucpy23miDcoquarumbarcsfunttri5gularc0p:cnmiconiung.itur:efftcicmr priamisauusbafiscnttctragonusmaioii tnagulartum cquilatcnis. IfiHiqucIibet carSenteopofita er oibus rriaguUribusqfunfatriagulan bafis fuerlfBaPvnitatg at bafis minoris artimus cll triagularts poflbafem maiorls.coiuncti tgif firiul pquinta buisfacifit tetragonu et fetius ontus 7fetius alren*’ fequente tetragcnfi. ct fic eouf® rQ ertrema tniras mioris Cii penultiroo alrcrlus rriagularf primB pofl vnitatc coficiar rcrragonfi.mitas igitur fi apponafoes aPtpfauf®tetragono8fupzapontoebabcbim<‘3billorcj tetragono cuif latus cftcqualeUterirria» gularis bsfis maioris ppramiPi8.at per fmlflam ev otb<’ ab vnitatc tetragoins eoportifur pyramis cuiuebafiseftqnaPrlguIsTconianctorflvltimustctragonus.dFctffi cftigirarqti proponebat. jo eSi quotlibcttnagularcs equales maltu coponafur: fici e? eis numeras ferratilis. CTflani qr equales erunt poterfit Pirectecotinucin altum latera vnios fuper latera alterius aptari applicart®:vtlaw8PefupraUtu8ab7pf:etferupraIarera8C7Cb:etlatu8gbfupraUtu3Pc:et IateragUetlzbfuptaIafcraprctfc:7itePeconfcqucnribusvf®aPVltmil5.et quoniaomniC Utera fum equaliaierit vt fupra pofira Pireere lateraqucfuprcmfi locum optinent ettrcmio cqualia fint et cquiPin3ti8.patetUa®perPiffinitionef3cromefreferr3ti[cquopptopoHrb3rur. «4- •f •«e •< 15 b it 3 h • g* •(? f • p•«e c • s« «b o• O O 0o oo0o00 eo oooOoooo •• ••• • oo0••• «;ooo oboo 0 0 CBmnie fciratiUa pyiamioc fuc baPia vito slriotiaBoit fupcr cam: priamtoc cuius balis cctragona cttibal! alteritts cquilatcra. CAatno€ptatriansuU nt>diatvt baUe inbon toUamr et o<CecGoo equaliamrrtan ^lorfi fecunoua pnamt^ia oerral^etoetertio teittue:(tc(]$ oeinceps.q: ferratilia vno altiot pfiao tnioe fucbaile pontuseftoe peniUtimotriieuUri&equaliC aufereC ruptenta vnitas step ppramlole tertflcflcqj tetua vlttmua triagularisfcrratiUs integer relinatur, at q: in baU ferratiliB fuit ablata b^lrii^iUrprratntoio.le>eo per fcciinoibo^ustfeconol triiguUrie reltctU dt larus iin omnibus fup<rlotlb’nmihterqulqjfi tot funt quotin lUo latereTnitattBKonllat latus ipfutncafcqnftiunila teribua fummJ quaozari pfiMre.p»tcrea qz tmd triangularis ferratilis remanet lamsfuS tminor fcqa?Bctinfequ^nbuarimllifermt«tlUirntfequ«ttsqnaotanlafcravCtf?ocpactoaOwtt3tevf(p quaozatop numerus ctcrefdt^tiergoTtlnfupremotriansularlreflbuanrmltaactqueenpzlmus cctragonus. ergo per ncrfitnloctauam bmus conftat rellaam cITe tetragonaprramtoe. et qzlatus balis eiufoeefl larus rrianguUn&equalium et latus baflepiramitis trilguleablateiofzconllatpr rBmit>€ relictam cITe pyramiDl bafts fcrrattliscqulUterSateplpfum ferratiU eamfupcr pyramlocm bafisfucaooerezquoo dlpzopoUmm. Ci0mni0 ferratilis ppamiDCfuc balis ouobus alno:: doe tnplus clTc p:obaf CSubflrabo eni ab (pfo ferratili cctremfi triangulartfi rdlnqueftR ferratilis tno alrtoz plramioc: qper pzeceoentf in ouas fecabtlls c(l piramiPas»)U3rfi vnaeft triangula ppramls balis ferratilis 7 alteratetragonica culus baiis tetragona cllet baft alterius c4latera.fent beceaoc tetragonicajiyra mis penzicclimanonam bulnn in ouasmrfus fecabilis e(V trilgulas pptamtues: quarS vna triagule pziorieflequaUsctalrerae8minor^i7umcflbaristri3gulepzioz{0. aootto igitur pzius oepeo triagu lartelbl badprtozis maguleppramiois equali an minozlUUa Ret alteri equalis^ficRIt tres cquales trtangnleprramtoesrorumferratilan conflituente6.conflatita(p totum buiufmooiferratiicm pyra mirituebons eife mplumzquoo efl ptopofimm. )t 51 o o 0 0 Clatcrcbalis Iriagularis oatoitriaplc ppamioia fup cam pftimtc fumma rcpirc. » ClScperfatur trifgularis lateris bati per ferta buluszin quclatus ipfuni ©ucatur tbabebimus fer ratil€eqoe altfi pyramloicui sMiccouos cquales triSgulares ;pucnt«<p ferratilis ouob^ pyramioc fuebailsaUtozcuiusfcrtlaparrcm accipctppzcccBftcbabccpyramioisfuB latus coilitutcfuntma atcpqC er tnagulan latens cati ma di oftmsfuis infcriozib® rriagulartb'’ coaceruafzqb c^ipofita. Cl^Tjamiois totius fumm5 rcpcnrc: cums balTs tetragona :ciufoemtp bafis latus 5+ oarum fuent. C3t pzcccoescocuit Oat# triagularis latere qb er tpfo tolbus fub eo co tentis ceaceruaf inuenire ita quo<5 et pfens nosinftruit cato tetraoni latere qb e% tpfo et omntbusfub co ptentie tetragonis coaccruaE rcpcrire.boc enteft tetragone pyramietsfumma reperirc quo» oftcettur.nam carolatere qocro per ferti bnlnseius triignlareminquevtinpzeeecrtifactSelicucolatus ptopofitdetpzoue nict ferratilis eqneatc? eius pyumis altus cui aooo Tn&conflmiU& triigniozum et venietferrati» tisvnopyzamiccfueba(l6 3itioi.cetracta igitur CpzeceCentS inuentabeflepyzamlceipertrircfinii pziin3bnn>o relinquetur tetragona pmmiseuiufquiCEbarts tetragona eft att? triangulari equila tera quefc; eft fummapetita et piopofitum. eSiouo fcrranlcB guoaimbafcfl triagularcs pjopfmi cpflant :ci quoium altitubo ,5 ,vnaict lateri maioiis equa fiicriccomimganmricffieictnr eubus. •• • CiRa**! inlpfis cqualiter fcj faficum numerfi lateris maioiis ferratilis funt hnultipUcatc i balisvnius et bafls alterius eSiuncte per quinta boiud terragonfi eifirifit cuius latus lateri maiozis efiequalctfecGcus triigulusvninefecunco alterius eoiunctuspereanceterragonQ conlimilE.r ita •cofequenter vt tot tetragoni cquales fuperponamr fibic^ inuiee aprEtur quot in latere funt cnitates ••atcnmcrillisligurantconfKturabafimquaczangulababensinlongumlarum etaltum cqualiter 0••ciitenfa per cilnnitlonem conducitor effe cubus. C0icuiUbet cubo aomngamrbafis fua ci triagulans bafi fuc c(?uflatcm6:cffidctui' 5^ mplus fuc pTiamioi. .^ CSnnio ouos triagularcs piorimos quozS maiozfit equiiaterus tetragono cublamgnan ct confli tuo per triceflmaboiusfupcreosfcrraKlcs quota altftuoontequalisUtcribafis maiozis qui cuo ferratiles coiunctiperbcccentlcquabBror tubo afngnato.iqtmalottllopferrariUaeftcq altus fue pyzamlcizotiplo ergo fuc balis nblacottoper trieefimafecunoa bui» fictpyzamiei fuebafis triplus, at qz alccrfcrratilis eil alteri eque altus zeritvno alno: fuapyzamice.baft igituripTifemel accita fiet ricefuepyramioiiriplas.atbafisfua cum baflaltcrius per quinta buiusfaciettetragona cubu qnarefipzecictocuboaccaturbafisfuadlbafimaiozisferratiUsqucefttrllguUrisbaflfuc equi» UteruswffideturnumcruspiramlDifuctripluszqberat oemonflranca. ClSontpzeterca»llfine quevtferrarilespcrtricefunabHlueftdlcfozmabfiturctquarfifuromacucencobaltmin coita cos gnofdturquc qjfaclIiafBtautotemtfit.funtetianumeriflrcBlarcsquiptopzqfemBpzincIpOreuer ftcnefozni«ur.etcac^rati0Defpb«rcfpbcrid<p<ublvtvlgittttqulnqjetiri^mafe):qui8raoicibuB Viin ^nafcfitur nomen tircuUres Mcutur tcewom vlgfnrttiuim^Bflhtmrfta eecew 1fef fppere fpl;cricn?5 cubi nounf qui arafiabus quinaria tfenaria infuM drenUrea qaaoraroo ouctis ^rearor.Seo quia bi numeri apaucis tpftimtis numeris nafcfimr quinario fes et fenario TBcmoftrartonlsolfricuUatcm non requtrannaut n&mcii cctena quaorons tcutns nomum obfer uentibofsnipUozcm ocrermiarioem autoi non ab re omifiiTe viociur. CfOctaul elcmcntoium Sritbmcticca Joioani lints. iSualitas cll mcqualitotia pzidpium .Jncqualitatfo qutncs funt rpedeat eaut ent maio! terminotum piinct rainotcm aliquotica vt iticbil fuplliuca PidE rauluplcr ppotno.aut femd tm aduaaltqna partan: afuppaiiicu» laris appellatur, aut fcmcl tdus aliquot partcsid otdf fupparttens. aut pluries tdus aliqui partem:ci multipicp ruppartiaiIarisvocatur.am pluries tdus aliquot pattestet mulnpiejc fuppatncs nolatur. Cdfcmotis quot? ao maiote babi» tuoo tonocm fpedes cotinettque inter fc llcut ptiotes ftn ocnoianuum oiucriltatetu variatur.cSdimpartcmaluio mulnpiicarculltotam iUtuorumere partem: quota fuerit que mulnplicat.c^uoodtnepaliUB mulupliceriiUuojjoun oldmnsqb ita fe babdao mulriplicammtricut multiplicans ao vult, cmioi blof alitjD per aUuotdtm illuo fumitur muloens qS per iocm multiplicatum fummamoluifi rdKluat. c^ncqualtum numerotum maious ao mmotem ea dltqucpattis ab ipo minotc bc nommatetao partem amaiore oidam .ppottio. lESinta tbnumcriincquAlimiamaioi/ctb mino:. (Intcpbepartes ab ipi8benoIatebfc3benoists abbeteocnomiaMaba.cico catiem^)poui6cmccaabbct046e.ponoen»vnfitnteractbqSntc:T vnumintcroctcqtifttf.et 4a quejipoitioaao vnum eaevnhisaopartem ab at>enoiaram:atea^ ;ppo:t(ofao e.ctque4»po:tiocvniu8at»b:caeftpaittsocnolateabbaovnum.aicacfti>aDf.i^mr pet ricefimafclSam fecuoi t cqu5 pzopottioalitatein Inwrectam que 4>po:tio aao bea ^oaO cipti» fc5 3minoieoenoiare/ao partem amaioreoenoi3t3m.quoo efl pwpofltam. C'8^altc$ quotiilict tti pdited equsica lESinipartes oarea/bcetnumeri eas oenoiantesb/e^et^^iitfiiit rebucereab e in Cies eiulW t>enotaridi6at(pequale8.fumopervice(Imaquitam terttf mimum n£emmqn2nimiersRtoeft4flt0 numercnpipmftnb/bl;et Atmparsoenoia» abg.erttenl per p:eceoe8remst0aoo«Maaorout» nficratgfmbsisitur^ mnumerabit afmb-stpctMnoemvtgaoertiabaom^afifninnerataftnk iSiturtmnumcrabitbfmh.itloeintmnumcrabitcftnlpcriocm.erfinpreNietepsrtestneinptcs equalesfcjainpartcs oictasabg^birtbincafoemfmk/etcineafbgpanesabgoictasfctfm U atfiabcftnguIenonnntparsifcobntpatrcBtcpartcsioemeuenietverutqfpartesoicteabgmb kI.TOUCCOO amb^ucnlut partes aoicrc ab g.cttmceDoblnk4nienifit parte® bab wP^nfierooicre et ouccnoocinl^jueniurrcouctcpartescoicteab g.quasfivnacolligi® coUectumoftenoUfummam partium ab cao canocm ociioiattoncm fc$ greoucta?» Clflumerum cum parte vel partibus batumtiu partes nmllcs rcouccre. CSitanumcruci-rbicopartesoatewtiubcaf ptimoatbmparteelTEcare&ocojeJntcoBnnHyum ^ ocnoianrcjpartcm bina.ct 4ab eflpars vnius tvnum pars lotiusatodtoiataabaiCTgoBvicenma terrtflmpnubeflparaaocnolataanuero qulflteyouctuDcnotann6infefQOenoianne b:maj>eno . minatemvniusipartisfc; a.oucocrgobmaT^uccf numerusoenoianBlUamparttmnBOtltmra onem muItipUcane afin illam partem.capio ergo (Ifes tlU porri fmg»tnictura nnmmimqui emoee fifc® parte® acuiaooo parcetn btUt agjircgatus efactficp apparebit piopoiltum»aa, nliueamur a nunicrunictpartcsbcTOinp(irc9partc8reouccre:rcoucopmoEpieccwembcta inpuEesBtes. * lOcmoettpriusOHcoHumcrumoenelantcminact^menietnficruseenotans illampartan qi^vt piiug compleo tcui aooo nflerum partium a» eanocm ocnoiationcm reouctayiet effteit politum. C^uoD fit cc Ducm Dtia? partium vnius in aliamic pars Dcnowta ab numero qui fitCiCDucfuouo^rnumero^vniusinalmmillaspartcsocnoiantium. CSintatbouepartcsiTnOericasocnoiantcfifintctoiteecinofiatc^tptrsoenoutaab enif. oicofe(rcpflrtcm4)ouctamc:cainb.critcnicj:oiffimtioevtvnumaoc:itaoaoc.|^coivtTOamaoc ^ itaaao vnum.crgovtaaovnum:it3 09oe.feoBPrimambuiu8'rtoe'De:itaf aeb.atBouTminoem ^ vtaaovnum:itaj)Ouctum eeamb aob.atfitafcbabetaobrfft igitur fBoUfinirioemqc fiteiain b.quoo cUptopofitum» CSiplura quotlibct in aliqo oucanf:tm ^oucif cptum fi qClibct co? oucaj in ill6» CIlt'ecocctmcp:iminnocflfcocoioi.iTtn6cru9accoUectu8Cj:abc4mu|tiplicctoftucofitnaffU8 * riucpevelBtes tsueniat c.cieeouctuaintifiatf/etbinofiatg/etcinofUib-^wofgbfOTmimee 5’mme.nameicoifiiniri6ceritvnumaDa<etaobetaoc;tan5’oaofictaogtaob.crgoEwceMn«m ternam fcc^Dtvnumao 3biciltcutoaof gtb*st BOiffmitionemflcutvnumaoa becitaoaoe. riintigirurfgiblimulq’mme.quoocflp:opofitum X itf I? <44I i? f 984;JI ^5rII lo I& II 1 Compofirerncrisnim* i8 lo 10 I (Compofttccttrcmarut e55? 5! 54C4 1 5bj' 1 f,, c5 e7I; 4 9>a5 59i!j » n €Si jjpofitio ab vmtatcquabrfltiatnucri cis vno minores rumatiir: fupparndarcs 40 ab tllis Dicic:p2oporiionumab vrtitate erunt oilTcrciuic. CSirvnitaeitt numeri pofl ipam abc:ct quacratl ccp eef:ct minores cisvno gIj ft.blco fucparti» culares Dtnomlaiae ab g/b/hqucfmrDat>g.c3S)l?.ctfaol?:cfrct)ifferct(a8 4;pcitionum ab mirate Tumptapvt 0ab gefle piopoitiencm quaabuncat oupla fup fcfqualteram et c ao bcfTcqua abutifat rcfqualrcra fup fdqiitftiamibfiti^ min mcoum pfcquctcr.lHam pprimam quin otffcri tiaqiia abfi: eat aaomitate.et bao nef! ^jporno qfit ci ^uais ainfc tvntiatts in brfco ain fc facit eqnqnlcc bcfleiuaqueeramsimUaambpcrfc^amiecdmanonampjtmlq-tomg.lgioirjiportlooaogeft eilfercita ^Tporcioiiis aae viittaicm ib3?’a-et i?8c roncoftceciurjiportiocm cael? icITceiffcrcciam ;ppo:tiot6 bao acr ca» b.ct ua rc qmbufiibct confcquetcr.quoo cU propofiluin. CSioroinetur pntiuc multiplices tfupparticularcsrci itvt citremaru multipliaum 4« qudibcr oiic coniuctc mmus fint ^nicoia ouplicatavcl oucmcoic coniuctcrruppar^ tiaibrcsvcro maiores. CSintprimc pnnuc multiplices abcteico ccpofitam epateminotem clTcpjopoitioem^^jportio bDuplicata-nam quia !>cnomm3rc6 ptinuc cqualiter fc cpcecCtrcrgo per ftbam primi ocnctons bcft mcoietas cenoian tiu atcrimuirumptoB.ct peeoinanenam ciure^ocnotans binfc maiorem ^011 = rtf numerum ^ain c-aip octaiiam buius bin fc 4J5uciccqualc;<ompofitc cj: jjporttonc beuplicata que per vicrfimSfcSain l?uiu© erit muliiplep.er tenclans ain ecnolantem cr^ieucit benoiantem per tcem mrtipUcts ep atc. igt lur nmltipkp er bouplicarara maiotennero oaiolata 9multiplcr opoii cacractcmaiorcninuItiplicceoporu3cra‘tc.qtienpr:mum.Sint(ccijcoDcfptinucriimprcfups particulares qua? sfiimarimariciffcraiccapgTb.bicoyPportiocsotffimul maiorem pUituerc p:oporucn(m 9>fit ;ppo:tioccupltrata.namq: per preccrctcm partes gbcnotanfammontuniicio cf partes tiQbmaie«:nama;numeri6cctfnuoruqua?rafop vno mino:ibUB.crso per primembums ;ppo:tio gmaior efl iportice b*at t> iffutit ranqeg-r f.nam 0 per fe tm cnq'tum cg.et cbio tanqi e fccb-« monftrata memaior b.igif pcrqnintamquiiin crutocf maius q’Ct>uplicata.ctbocniooo facilciocmp vtraipparrcoilcnrssficucponaiurmcricajporriocs.fifcpcoduomirpropofirun!. C0nmparcspoftfa'n3rium?{iniicfumpti:apmTcrcopoftbinarmmnamralircric 4i orrmatesccmparcruricyitunrruppaincnfcopiimiaocnoianonccrcfccnics. CfiJiarabc numen roir binarii. meetinucfrirptirctbcffutrpiuotinucimrarcppoft tcni-irnim. t'ifot>a03)'ctcat' bi ttfao cfupperiiifcsoiHiuaccncfoiiccercfcftcc.ncmcu eiffcrctia impaifum fit binariae 5arci tfop ab inanuni quimincr cftcoTtio.igui;r ocoacftfucpartiis.ctitcireumcifs fcreiia imparium fit,bmariusictnijcrep ab TmratcPifpclitcEttffcrctiafitviiifaoracoctcfupb vno plueq’pfupa.fc&cfqrc/c/frm)rfcr!t»iucimrarcs:aeccfffnpc3Cbucvnopius§'efHpb.crcfeuHt igiruroiifcrcnticimpariumrupnucrosab viuratcfumptou flcufripfinncriiiragcotinncptcecocs cliclfffrcttaimparisfupftQucicm.feo numen vniiaictinarcsfunrptrafcprimuigiturcifreretiaq cflnnmerusprccccetisabfcquctcm inimCTopabvnifaictifpofiroprnoneftpars fcqucns.eftiginir Pprimam pmiciusptes.cntigifjrpcrtioc ab bfupparticns/^tfaocfupparrice.erqzrue partes fm numerorr vn/tate bifpofitcp qti tacem crefcuntrciQt «eap pt mue crcfcCTcs benotattonesnmo ^quia oenotanfanamcrisptrafepriiinaptinue^ccpcntoiuerfefupparticntifi fpeciC8.qdcftpropofiimn. eSiconipofirus cx primo pari 7impai irqui eje reliquis imparibus cipcm ametis 4? aggregatur:ao numeros aouobus tufpofitos comparetur: ^meniet oiucrfc multipli CCS ruppariicularcs fmnucros imilriplicium zocnoiationcs partium fUies babetes^ Sint abcbnSeriabinaiio ptinuefuccebetcsrnntcp ei fi gj bnficri aternario ofeqnctcr impares fltipkccmpoITtusciratcprimoparitimpan.ctlecmpofltusctfTfe.ctmeopofiiiis epg-r I.ctncp btmjbicofc;l/m;ncoBato8ab8ib(c;b;faccrcmuUipliccsfuBparficularcsfmniicrummuItiplicifi vflfcs partium bcnotariones.nam kpponif et 3 1 cbtnario ctcrnario.at eabbit fup apartem vnam ab ipifo benotatam fcj vnam niebiam fai febam qfi ibcmvaleatrcrgo keompatus ab abis fpm ptinet ereiuemebtam etabipm cflpzimusmulriplebfuBparttcuIario.rcb-zBbccimanonamrepnmlfccpo nirurcxaetb-etge);bctc.ctborcetb.ctperrcptim3rcptiquabratucuiuflibet obbitfup quabraifi ^rtimiqberipaseoponifvtquebratumbfupquabratumar^iLmqbeoponif eva 7b.crquaer3ru cfup quabrammb ^tum cSpoitif et btc.et ita bcinceps.et quia k abbit iup quabratu avntratem: I fimiliccr abbet fug quobratum bret mfiip qnacratum cret nfiipcr quabratum bvnitarem.quare fins giili eoEfmgulos illo? cotmentfmfeet partem econeeptioem ab ipis bictam.ernt igitur pfeqiiercr fmtpros;nuiitiplicc6fuBparticularr8.quoocftpropeirituitt. eSiDUO pmi impares piugatif:copofitocp reliqui ipares appliceturtcrit vt ppoz-^ 44 riones coporitoi; ao nucros poft bmanum furfiptos fmr diuctIc multiplices fuppars tientes zab coocm numero zpartes numeratas zmultiplices Denotatas babcics. esint nuo primi ipares a7b:« copofit« Ut cjRntc^ rcUq ipares b efqni aonafppoiito erfinteg ag grcgatig!b(kietl/m(n/o;iiumcripiinucpoftbin3naffiptl.0icocabl/ctsaom/ctbab n/ctkabo biucrfas e^ multipUcesfHEpsrtictcs xfingulas ab conem numeromuUipltces bcnoiatas xpartes Kunuraras.iia bacsit fuper abituriSurfioc bis continet alUi fliperabb^s binarifi. feo binanuA cfl VBo mlnotaicrgocll cius toQb£parte8.<;f(igiri<r caca pi!mamu(ii{»{(trupcrp4mts: quare taol. cttactlfuntioeni nuincros.9temqtmirunicftcab9ttbiiiariBfupcreoqb>unrlnnArtoins.ctb(> narius tnm per nona ptimi facit ctrrgoc bis corinetm.rcbct pctbccunjnonamfeptiimbcdllateicl ermiTffcdiraterccto.ergogcbnnettcrm.tinfupcrltresfcs ciuspartes.cruqjgaomfecfibamnJ riplej:fuEparticn8.ctbun£innioPfipiobabiolinncoftirufrcg.tpcrMcinMnoiia(cpiuminnconlh tucre f.1 bcotinerc niccunou m.ct infue mquattuoi fcj parres n.ct ita coftqnaitfT inuenies ituiltl pUeium OenoUrioncs et ab cvbc numero fuEparttcntiu partes numeratasiqti cft ptopofitfi. 45 CSi DUC oiucrfc rupcrparncularce comuganiur compoltta aur crji ouph aut fuper particulano aut fuperpament». IHiSinc abeo efquorlibct per ototne rupparcicularcstcico q> (i oue quelibet fimul iunganf contpo rtuinelTeautoupljautrupcrparricuUrc autrucparrtercm.nanirtAetbrcfqualm^ ctTefquuertu cSlungis copofi ta cntoupU.nS loiintmta t'upla:uiucui0 per ptima quinti ferquirertij jipottionem qua abunbatbupla fuper (crquaUera.conitimitigimrbueptime bupU. feo licum a aut cauto aute aut f cdiuntenetqt per rrucfimaicptinu buius queUbet eit mmoib c^fiia ^poni» eru tiia btipla mfnot et ibeono muUiplcs;. cfl cifi bupla per eanoe tricefimarqutuna buius multi plicidnunimatigif copofita^pottio aut erufucpariUulans aut fuperpariusiquob efl piopofuu. +tf €0upcrp3rticuUrcm inucnirctcum quaquotlibci fupcrpamcuUrium qucltbct:fui pcrparncularcm componant. CISint abcquotlibet numeri ab vnitate a>po2tionalc8:et (It bpdmus as aqui Ibe etii per viceflmS fcpcima quarti erit ptimueabbccctcoiungamrociulibetcotum ttptoucniSt compofiitef6.cruq; f >er nona tertii quiUbeteotu ab quelibet eota et quoflibct altos lUius oibiiiis ^ipomonalia ptimus* umo itera edtinue 4>pouionaie poflptimosquifltbetpervicerimaqumratcrtgminimitqucnumc* rant bcfgqui iit l: qui pcrricefimaf^timi tertq ent ptiinus ab b>lit itepervlnina tertii luiuUtplee btmiiiulriplcclt.abcarcplfuperm^tum cflb.beinbeabDolIgtUatiaDlnumerosabcecptoueiuat copofieinpqimanlfcnii cftleiTcfuperparticulareabni.nacirupcrabbitfuapartemfcs b;feo «net fupcrabbif biaquobcfte.ctpeifupcraboltbttbqnooeilf.tqeifupcraobubttcquobeftg.tpcr viccfiniStertiam ptimt cfgquilibet cit pars miqtdl pars U.igitur ri pqpariter et Itunt fupparticui Ures ab m.conilarc ergo poteilptopofitum. 4.7 CBomrupcrpattioilanitquotlibctaUao inquirere quarum quelibet cum caDcmifu^ pcrpaitientcm cffiaaf. ClSitabenominaspartem oaterupparticuIaris.niino ab vnitate cotfttueatpottionales quota cpfe fitvnus quifintabc-bico aipotrion^fupetpartli^rf bktiabacumruppartiatl^etrluparsbe! nomiitaf ab b vel ab ccsnfhm«e fupcrpamenc&quUn ptefens ptopooicinuention&namcoponta ep a>po2tloncfupparticulari renofata ab aet benoWaab bb^ibct tn beholatione fnivnfi etparres i^ btcurur ab 3T ab b per octauSbuiiis. qqutbf paries per quanS^uius renominaturab eo qg ftt cc a in bqui fit bfes ille per becimacertia buiusno confhtuunt part&ergo cO ptopottioe^ioqia et bemo flrationcpcnultimenofitmultiplccnenfuBpartuuUnstqrpartescompofitenoncoftituuntparrf vna-relinquif ergo compofirl elTe fupcrpartiente: tita be qualibn alia. 48 C«0mnio ruppaniculario cu qualibet fuepamctcicoftituit ppo?ti'onc mpla mincte 4niRam oitpla cliferqu altera que per mceilniafcptiml buius edmacU ruperpamcuUnii pponimt per cotrclarili ricefimctertic l?uiue tripla: feb ots piopojtio fucpartiens dl minot ^ipwrione oapU Igitur queeuep a>potrio fuperpartifs cu manmarupparticuUn aut minoti fupparncnlari pquuta quinti collituet ptopottionc tripla mlnotfiquob dlptopoiitfi. CData piopoitione que tripla mmoz fittin fuBparti'culare zfupparticnte partiri. eSitpumo fUpparticularie quecunip bata quebocmobo oiutm petatur^apto atb mimos terlos aquibc minore t b malote inter quos cofiftat-qui per triecftmSfepruna buius erunt numeri tn conti naarericnuntero9rumpti:folacpTnitaiebin3te8.ntq$ccopofiio«(i:d8cuiabb^uraet4nieniatb: quequtbemaiot vmtatclitctbucaturaetbinbeTp>oueni3cfctg.eHtoipcroctau3feefibijtpomo caofea qucbaba.^tcbucoameT^ueniatbtabDctqjbtuixffPf'’"*"®?»"’^®’^^'"^^'”®^^® r.n3 apofirus dl numerare ffecunbfi otigitur l? ab fdl iipotfio rupparttculans-^rem qt beft ttiaiot avnitatcperbccimapjimigabbit fuper f5’tumeflo.ergo gabcu fuper bminus bquantiicft a; teb fubftr3Ctoaaboman«c.igiturgabbltfupcrbq«wnifi^C'*P^'’®"“'’’^«f"^® terrqceflpzimus 00 0.crlt itaipg 30 1? 4>po«io fupcrparticsiquareptopojtio fupparrtcuUris affignata quedlg ao f In ouas biuifa cftifcy bab ffupparticularew gao bruperparrtente;qnoo eft pwpofira quo ao boc. SecunoofitfuperpartifsinterartbminimoatcrminosifitcbbmaiotteTminusctbucanTbinfect inaetpiouenlitctb.eritqjperfeprtmSfecunol^potfiocaooeafuBparrus qiiebab3.5temabbo a30 0et pjoucniate-pcroeeunapjimi anumerabit efccunbfinumcrlivnomalot^b.quarceabbefl fupcrparticuUrie-CT quia per oecimam tenfl ccftptimus ao b:crgo perwoeomadufocrn cfl «lain ptimue ao c.« cuniotlTercnttacabcnon fiivmtasfeb vnltate maiotiergo non erit fuBparticulans. delinquitur igitur caocpiopottioneciTefuperpartiacmiqnarcfuBparncsairiBnatacaoboiuifj efl iftbuastfcj eab 0fuperpamcuUrf acao efuperparttetemqufiaonwoS erat pjopofttu.Zcmo fit cI 54- q P n— — •R 1155 * b—^Itf gII c8 f7b^ PJI t7 tS tf S iCompoiUc 98e— 8 e8gi 1 D7b1s c5fi 4 3Jbj I5 t8 i5 3jbtf e8f5 a—ib1 c504 I (ntcr anbpfcttio bupU ()ue qtvt bemofh-atum cfl (n ()uat>;a0cfiniaqulra l^ufue conftat nrcfqual tcra tftfcimtmta ftibitral^f per ptimiqulnti^^onio rupparnculane cao »«luccficp mtnoz f^qui: tmtaa,ppo^oneoob:(ucpreat>uaeaof4uaab6bataaobrupercaoo.‘tq:maiotrii^po}rtone rcfqaalti^ que cft marUruperparticularifi necp dl multiplo:;: at ppottio aao bdV niinima muUu pU(i6 qtunj refolititar in mDlhpUcca.rrtlquitur er^o ^^potrlonee ao feflef^uppartid^. rt qt ce& btct&f(SlUtuiint4>po»ioniaa9bconlUtjipotrioti£ouplam inrupparrUularctfuperparttdcm cfrcDiutTaRuCaartofit intrr actbqodibet4>pomoa&upU ac tripla cdlhtucaifircptniercet opto pottio cSpofttaet KtpU trefqultertia li ptopottlo aab beflminot ;^o:tionc cao D.n-go Tubllrairra ^portione refqaUcrriA a^poztione aaob remanebit ptopoitio minotouplaircD niafotfefqualtcra. quareilla refiouamtfupfii)8rttre:tricMo«toflaobclurtc5pofifae>:fuBparricH lari vtrefqterria et rucpartteti tc etqt in caa oiuiftbilie.i it aat> bfuequalte cao baut maio2:qt^o;no acolra bupU {ti;pponiM<e aBO b:ent minat rcfqnaltera c(> oupla ifcrqualrera per cotrelanQ vtccTimctrnic bui* confhtuit rrlpU.ct fcfqualtera aoMta refqualtcre mmnsc^tuat ^cupla cS rcTquirertia cp rcfqual teraetcueretquitmiepluBflnt^refqoaltcrartrquitmietfefquiocteua.atpcr ptlmS quinti beco: (pioTcfimr cquiwl coplate rcfqualtercit illi cuplert reTqiitrerne.ergo Tubllracta fefqualtcra ce pto: p«tt5ea ac bremanebit j>pottiominei cuplarfco tmaio: fefqo8ltera.quarefuBpartif8.eflrct itai® pzapottio Bacb in ^ottionlfuepanicularc rcfqualrcra et Tupcrpartidcm quomoco ptopofitG recabilta: quee eftpt^ofitum* 18C^ftawtna ruperpartientj mucnirctqtic cy ouabne rupparticulanbu© poudmr. ?+eSitaac b4>potrfo fefqoaltera et cac cjipctrio fcfquiquartatnfip ees f jipottio ec bie eccofita 01cico ptopottumi eaC feffc majrimS fiiBpamctem qoe et cuabue fuBparttemaribus ^ucitur. na qt Cfipla et qoacmgcfimaquita bulus ptoouaFe% refqualtera tfeTquifcia quefunt matteruBpertteu-^ Ure«:ergo.pp«ttoe8Cfeftminoz4>po:ttoneOopla.fcctcacamaiozeftfcfquaUcra:q’ctcacum aU lerat^p^Mnt^gocacfeil^ozttofoBparztenatrcctqueUbetminozfuppartinilariecfiptcpote tiMuaaobfef<biaUeramutcaccfefqutquarta6qumtaqninnininoz<4tpotnon€conituct&cacff e(l igitur eac fmatinta ruBpartifa et cuebueruBparzicularib^ ^peuctatquoe cftgipofitutti. CSupcrparticniSmucnirct que mDuas rupcrparricularcs bmioi non polTir. vnoc t,j, pfltctquaiibctfupcfpartientcaliam cffcmaiotcm. '5 eSitbac amattma,ppo:tio fuBparnene per ^ecente cognitaiquect Cuubus fuBParticularibue ''14- gmuaaftteclfferetiabacafltamanifciWicnafncmaio.-ccruiequiCcparrbue.capioirenumcrfi cvno mino:eb:t tune^tpomo aac c:qj amaiczc parte ccnciatur maioz cflS- pzopoznc bac c.acoo ergoecfibetftatctmamfcflfieftcaobc/TernEFarrimt^cumcotincarbtccitiOBtcs-etpcrfecunca parte eeeimequtte fecunci crlr maio: ^portio cac b^bac a. ct qt ^ipozno baca ciutetHn matiae ruBpartienIaree conihtuftce fupparftctcmrt ct piecccenh cognofcif et fucpartiene cac bfit maioz fupCTpartffteb ac e. no oiuicet ergo fuBparriena eae btn cuae minozea 5b ac 3:nc«? in ealbcm iti quaaciuicifbaca.relmquifergo in nullas cuaeniBparticulareefuperparrictfeacbpolTcciuici quocellpzoporitft.£tco:relarl&ctmococcm6nrarionl0cognofcif. CioolamulnpIicdquaUbct non poiclt fumi mtno: :ct foU fupcrpamculan quabbet non poteft fumi rnoio?. ClbicopzimoqualibetmuItlplicepo(rcruni{maioiS.n3pertr(cenmarepHmabuiu8cea ntimmao Bnirat^pparaticSiHfuantmuitipUaeettltcmaioteequitoabvnitateremonotce': fenee niieros in bnenfum euactMgiF qualibet multiplice potefl fumt mato:. SeCo Cicc q? no qualibet mulriplire p^efumi tninot.nSpereloem quito numenie «pmqulozvnitarefuerttiianro ille ac Tnirarcniinou' eSfhcutt muUiplic&at cefdccnco no abirur in inftnitG. non igtf qualibet multiplice eft care mino:e ^in aliis minime qn^acmocB monflrabif accicetXertio cico qualibet fupparticuUricfrcnimoz^ mlparsperpzimibninaqueaminozinumerocenoUturmalot^etquItoamaiotitninoz.etqz fe riesnumerotfi in immirfifuccrdcitmocabirar para aceo paruaquaminc: cari non pofTit. qualibet igimra8pacticularipotencarimino:4^artocicondqualibcrfnBpartiniIarieircmaioz€.na5 pu que anumero .pptaquioze vnitari cenomlnaf malo: cfhi q: no contigit numera in infinfrfi cecrefc^ cabif ergo tmic£rnBpanicaIariamatima.arTrriuffS oppofiractiamondameftinmultiplidbua acdcebat4Bnbocico quabbet fuepartiente efle maioze.boc et cozrelario pceCentia notS eftSecto cico omni ruBpaitienteeircminoze.nam quaefiep catav^bi canfa ru&bipartl?t« tertiaa fumo alia Btotici partibuamenominatia amaiotenumero qu£no numerer: vt TuBbiparttentl quttaa qqutcf q: 8toticc minozlbua Btibua cenominaftneqjptunbua erit mino: fuBpartiente eata.qua fuebipar timtequitaaruBbipamenareptimaaminoteiletbacruBbipartienenonaazriccpcorequenffmcro mentum numero? infinit8:quoo eft^^ofiratn. “Ip-oreft tiCTlrimfi ollcnoi et tertis parte pzcfaitier aomuantequaCMgefima nona buiua. C^urfibct DUC ruppamentee comucte: p:opo:n‘one quaD:upIa nnno:e conlhtucf. 55 hl<rS(t mter aet bpzopoztio Tuperpardes: iricem inter btc. cico p:opo:ftoti?a ac bct bac cITmpl: ^1mtno!«quaczupla conftimere ptopoztfonem. rtam adl mlnua cupio b: et bfimilirermfnua cupto c 'ergoa ac cperquUitam quinti efl mino: p:opo:tioneqnactopla:qucperoctauam buiua et cuabua cuplia confUttcilergo clarum p:opolItuffi. Vim .4 CD3i5p:opo:nonci}ucqu30mpIammo:fitinDU33 rupcrpamcntwoiuiDcrc. CSt^tt'atr^lpo^tto^{flmlnimifc^^1^ntact 3b:(^utconiuncfifac^4^t^;lTtl^ema^o^^^loqu^ct^n» tuncti faciant ftcut aooatur cquate ».nt($ totus fg:ouutur<9 3mfgct jentatb:t ct abin efioem fiatIjUl.patct tgitur g» cxbmfgfit Ijbetcibmgfitli. quare cjrbmffttl. fttltnntmplusabatt ci;ciiig(iattnit:rir(¥ncqualt6E:.pcro<Tau3mfccuncicrirp2opo:tlobI<Ua7ibtAn73ba&fl.iqau .1 bper l;ppoil?d?m cft minue 5‘quastuplu a:cfl ergo bminue ^tnptuma.quarc minue §• c.i quta gcft cquale 0ct ncqualte g:con(ta tmicITccqualem vcl multiplicent e» eper ouooecimi p2imt.3ce(n qtoetcfuiitcontrarepnmitcrgo per nonam reriflfcrir at» viruniyptlmue.ct per can1*2 fctgfljnul aoi>pjimu8.ctqittellmtnotDia»tpfum pnmuepcrcanoenouamtquoo fiteratnfgfcjbnotinu merabifurper pccimaciufccm ab o.ctqtrnumcratU tono numerat btergo l: per viceftmaterrlam pttntinoniimerabitb.ctqiitafgmainecrt^’ rnplum o;ergomalu8cntbq’n:matuecrgo 5’b.mi ergo hpartes b.ttcaocm raitcncqtabcnminottpttmiieabfnomimcrabttfquoofiictabinfg: fc5 bhl.ctqjfnumerati: ergo Ino numerabit bl:l:qu9rencqtblt«tqt per notii piimildlqoltnm 1:biectb.tncftcqualieltctmcqualiBtcI mciototmatotbctbmaiotmtiicritbUmflto: l-cntigr lpartee eiue.conftat ttacp bUjao beHe fuperparrien t^ tb bla® bb«iam effcfuppartlentftinqoae oiuifacflpjopomoquccftintcraetabifactumt^clTcptopoRtum. ^5 C(0mni0 muUiplcf fupparticulans aoDit fuper (Tniilc5 multiplicc^fupcrparricularc anumero qui Rc cpoiictu numen muUiplicio mnumcril pamsocnommara. Cl^it b30 ap:opo:rio multipUr rupcrparricularie: fitip cao 3Rmilie multiple^:. Oleo ptoponjon2 bao aaoocrcfuper p;opo:tionem caoafupparticularcnvocnotmnatam anumero qui fit ejoucto numeri mulripUcie in numcriioenominanre partis ruperparticularis. SStrcntoquoabunoatbfuB cet ocnominas multiplicis fit etoenommansruperparruulariefitftmanifcrnicRergoqTacrutnc fcomofi e: 10iii afccuitoum f. fiat ergo gee ein f: cnt<? per siccfiniatcrtiam p:imi 0in cfecunoQ g. quare bcontinebit cet partem ocnominatl ab g. continet igitur p2opo:tio bao 3ptopotttoncfupcrs particularem ociiominata ab gnumero qulfitepoiicrucoenommlttemuUiplictslnf numerSoeno miitantem partte ruperparttculerie: quob eflptopofitum. 10 ij I- tgmn 5tf S7 58 C^mm'0multiplcF fuperpames cuius mulfipliae numerueparteo numcmtiaoDil fuper ennoem multiplice :fupcrparricularc ab vna partium oaiominatS. f 1 01 C^Sitjtpoztlob aoa^^ttiomulrlpietrupparttcneirsti? 0numerus oenomlnaemulbpIwWm bSc— 8* qucfitfumptusnumeroopamfibaoa.oico 4>po2tionebaoaaooerefnpernmtl2inultiPu<*f^er particulare oenominata abvna Ularumpartiu>Samocnfcaoanmil2muItipIicf.f qtbibaioteicc: fitfquoabunOBtbfupcrc.q:entfellparrcsafumptefcbmoetoenomlnate9ba:Tedlp8t^cbe!io minata abo-etefi c;:oln afiateeefina fiet c.cfl ergo fparscoenomtatasb e. continet i^turbaoa ptopo2fionem rupcrparticular2fupcr ^ipoztione cab aocnomiuati ab nta parttum ftzpeiiiartims: qucoeratocmonflranouni. C-afeulriplicc rupcrparticntc mucnirciquc fuper tota multiplice aoDatfupcrparticu larcmicu lint numeruo partiu rnumeruo ocnoians mulripliceoati comcnfurabilcs. |r'l^toponitbcc'-q’ftbanniimcr!quo:fivnusocnotatmultipltccct alter cft numerus pamfifimul multlpIicisfuBparrietiGtfmtaotmiictcomenfurabilceiiiucniremultlpUcIfuBparti^tem que fuper fimllcmmultiplice aooat fupparcicularc.Smtpmo anumerus ocnominis mulripUcI ibnnmcruB partifioaticomcnrurabilcs.fitcBbnumcrusfumptusrciSmaetperpcfoenflftrtuni erit,pponrutn. SintfcfioaTbcomcnfurabilesifcobfirparsacraoiuuatepccoentcBticefimitertiampiimicoclus oee cpofitri. Sit tertio apars bcicta ab c:fumo quclibci numcrfimaiezl a: qui tamennonutnerct quifito:ctoucocinoct 45ucniate.tqtcinafacitb:critcmaictb.et qzBreptimirccfioibapcncut aaoo «anonumerat o.igtturnetpbnumcrabite.oucoatneT 4mcniatf:cuiatiiiinctobfiatg.t>tco crgogabccfTefuBPflrticntcqucqucrlf.namgcotinctefccunoSsettnfuBbpartese.iqtbcnpars fmcmultipict fit aoc:T 8lncfadatb.aooct igitur gfuefmultiplicf sococnomtnarfi ab amumera bciiis parteiquoo cR jjpcfitinn.flQuarro fint arbcomenturabilcstfco neuter aitcru numerrt. tftt c macimus cos numcr3s:numcr«<B cnumerfi bfctJm 0. capio cqulcuncpnumcrfi maiozl c. qui rami cnoiuimerctiqucDucoinoetjmeniarfqiileritmatozbmcipabeo numerabinir.eicoCTgowmpze* ceoenti parte imeoiate moRratfi e(h ^zpozttonem fuBpartientI oenominata ab at a partibus ©Ictis ab ffumpris fecunca numerum btaoocre fuper multiplicem ©ictam ab atfupcrparttcularl.boc «in pzcccocnti parte mouRratur: fictp totum conllruitur ptopofiram. CJl^ulopUcefupcrparticntecuiua multiplicis numerus ctnumcr^parfcsnumcfas fiicrintacitiuicc pjinii: fupw firailc mulnplice;ruEparticntc aoocre nccdfc dl. <rSitanum«»mult(pItcis«bnumcruspartiiiruperpamlti8:finti? 3ibnamcriaoiJinkfpzlmi ^0muUiplicefuperpartlentl oenominata abartpartifibjfupafimaemulripliclaooere pzopoz=: rionefupcrpartientc.fitentcoenomingspartesbmamfeftOeftceflcmalozebznecpbipmnurnerarc cuco erao ain ctfiat 0per viccTimapumam tCTtfl b no numerabiio-atmo igitur bfuper 0et fiat c:q: cniecontinctcfccunott a,ct infupCT partes cfumptasrecunoBbdtpzopoztioeaocmnltiplerfupcr partiens oeiwiata ab aet partium b.feo tpzopottio 0ao ceft mulriptaolcrs ab 8:« caooit fuper 0 10+ ^1+4 e1« 15 fIt 15 9t49 i9 7 1? ? g M4 IJ numerfi bqoi monfh-afus eftptrtee b.ccndat itatp ^ipoirtcnS multiplici fuperpamcntS ciaS ab a: etpartlCbaopcrcfupcrflmlUmultiplicftvtjjponcbaf ^tp«ticncfuEparrtent«qdeftp:opoftt6. CSoIafupcrpamculariu fcfqualicramulapljccfupcrparticularcpjccrcat. 5? C£(loa>obrcfqualterartba9cHmiUrer.manirclme(l4>pottioniaat)ccomp0ritaefreecbuabue fcfqualtcrisnaconcinetbietuemeoleratiqucritoictbcotinetcTciuamePtcratiqucfitc.etquia per quinta fccnnci oe» «(kut bao c: ergo 5> continet etctua meeietate. bergo et efimul tantfi fimt $tomcctvnapa»&icraabc<c5tinctcrgoanunierficbie:etvnl cluapartc.tfacarineatcnoerotj ctfferenrtaaaocquarcbucWquaUerccoihruontmnltipUcefupCTparticularc. SccunOo oftcnblf noUaaUamfuBpamcuUrieuplicataconRitucremultipIicifupcrparticuIarc.ctfutoabc^poniof nalee alte fopparticularesivc ^poitio aabcilt ^pottio aao bcupltcata. acotinet bct partem cius qDcfit bu]ucquioi bis fumpta mtnuit atoto aut fcmcl fi cflet terr(a:aut bis fi quartaiaut plu rtes 9mino: pars fuerit, ct fimilif bcontinet cifimlli partem eius que Tit e. et q:o ab b ficut e aDcwrgooaeeftcutbabc.cotincTcrgoofcmclcrteiusTtiSfimilipartemiergooctcriniulminus funtJfc.quareaabcnoeftmultipierrcttctcfibtefmtcontraftpiinilvtfupponim^ioctcriniulper nonJ terti) eft primos a» c.fetj ct pet vnoctima eiufsi cetiam erit ptimus ao 9 ct cfimul. ergo oct c flmul no ruimerabiit c.eflergo aao c.ppo:tto fuppartiensiquob efl iPpofitum. C^ulnplcrppoino in aU4i cqualcpjjpojiioncp piftnbui non pot picr mftipliccp. eSit4>potfto 889 bmulriplec que 9iui9arur in quotlibrt ;ppo«ione8 equales qfTnt aa9 c: ca9 9 et9a9 b.qienimba9 9‘t9a9cetca9aruntcontinue4>po:tion8leseTbptinius per^fpotbeiTm nu merata vItunum:ergo per bcamitcrtiam quam bnumerabit ofecunuG.quare 1 9 numerabit cct c itibetn a.9iutra efl tgttura 89 bin ^poitioncs multiplices: quo9 int^diatur. CflulUfupparticulane ^potrio maliquot cqualce rpowoncs cft Diuifibilio. eSifaa® 0quellbet ^ipoirto fuperparricularisiquc 9iui9af fi pofllbllcell in 4ipo:tionc8 equales aa9b:b89 c«tc899.captoefgl?^>po:ticne8 in mtmmls terminisjipomonisaab b:et per equam ptopoztionalitatf que piopottio aa9 9ca d> eao b.teb acotinet 9et eiuspsrtn igitur ct ceorinebit bet cius part€ que fit litque 11 fuerit numerus cfl Unumerabit bterta c.crgo bet cno erunt minimi quo6 per quinrf quarn repugnat !)7pctbefl:.alloqu tnl: eflrnitas eft pars nenoiata ab b* 999o an b vnitatf: (Itcp copofirus i:enf:p quep:opo:tio aa9 oea Ian b. ergo per 9uo9ccima quarti quotmena ;ppomonale9interci9Ctinrer3er9:rori?gtlnterlctbfoIavnltatc9i(l3teo:qi5cflIpoiribilc.£]c bac tognofrifronD8inmuflcaqutcrfcr4octau3^po:rionenafcitur:in9uocquapart{nopoirc. C0upcrp3fticiitcp:opo:tioncqucouplicatjruepamcntcconllituat:inucnirc. CSumoqualibctp:opo:tioncqucinino:riteuplaqufperquinquagcfim.lquart5buiU8parriottii ' nuas fuperpartientes quam mmo: fi ouplctur clfidctur piopoitlo mtnoi oupUtquarequallbct mul tiplice.que qt per pieccocnre no erit fupcrparticularisalioquin fnperparricularis in 9uo cquafciaj retnnconftatcameirefuperparttcntctfactumipeflepiopofimm, C^feuItipUcc ruppaincularc qcp puab’ fupparticnbub cqualib” pponat: rcpcrirc. , esumofecunnfi erigentiS nerime ectaul quantatos aet bpter ptimos quoifi aa99at fuper buolfi bvnitarcmanifeftfi dl ^iportlenf a89 beflemuItipiicJfuperparticularf. at ejriUa 9ecfma facile co* gnofeea piopoztiong lateris aas larus bciTefuperparticntf.fumo ergofecunsfi erigentia coirciarff quartcreCTimc9{6ptopoJtionaleinteraftb:eritqipertert(3cluf9«fc)rtipjopoit!oaa9ct8it3’ o:o« poitioUteriiaao latus b.quarcconftatpiopoiriene multjplieefupcrparticularea aubcemponti ciTceirpiopottionibnsaabcetcabbuuabUBrcjcquaUbusruBpamcnribuetqtJcftptopofite. CSupcrpamcntcinucrare qiicbuplicataitiuliipliccfupcrpamcntcconHituat. i CSitaquilibeitmparetbptorimusparmaioiculusmebicfasIltc.capiofuperparrtcntfrumpfl aparribusbenofflinaHsabatndmeratisabcquamPupltcatSsicoefrepctitam.namfiillafnreou caturpiouenietnumenis et partes numerate aba Taqua9:ato equi ITrc:et9enomtnafcaqu89tato equi flt9.fe9 cum atefint contra repiimiaUoquina et bcomuntcarct.ergopenmscomatcriiiaell Piimu689e:tp<rn<maeiafi>emaeteflmulftiittpiimu8a9a.qu8rcaete(fmulpcr9n9ccimictuf9e nmt piiirms au 9. ergo partes numerateab aet cno numerabut9. quaregjponio fuprrpartife oue adlgnataerat bupltcara cSiHmit multipUcf ruperparticnte.ia[ fi bec semoftratio tibi fiibofcura vii 9crur ptobac tfequaibua setlararfon5 acdpe.qi cni in piefenrifupcrpartifs inuenta erat fuperbi' parties tertias qnesenominafab wio xDuabus tertiis.nuc ergo Denominatione fuperbfpamentifl tertias per octaua butus In Denominatione fuperbtparttetis tertias fcj ouceoo pnmoDuas tertiae in Duaa tcrfiao.er per quarta buluepJouenifit quatmo: none.fccfioo Duas tertias in vnQ tJiueniut tue tcrtie.tcrtio vnS in tuas tertias et iterum pioucnlQ ttue fcrtic.t quartovnC in vfium:a±iicnit vnumibabebirqj quartuo: nonas tuas et tuas tertias tvnfiifet Duc -t duc rertietlD fquattuoTtertie faaunmum mtcgru ctvnam rcrtiamiquoDquicl vnum alteri wit astitu conflimit binartum enma tertia efl tres noneiquos per fecuntSbufus factlccognofca8.quefiquitem tres none aliis quartuo' nontspjimotnueris aD9anturfiuntfcptcmnone.ip>atetif3t5p:op9rtfonSruperbfpartfent5tcrtiafl DupUeaum: confHmcrcouplamctfupeifepniparrtentenonBS quceftmulriplejifupcrpartlcnstvt pioponebat piopoUtio: quot eflpiopodtil Declaratum. I5 C4feulopIiccm ropcrparuoildrati jnudtigjrc: que mulriplicetu fupcrparticularctti Dupliaf.ip;coucflf. Ctapio anumerfi qu?canipqucm oufo in fe tprouentat bcuj aopo wiitatftfucp totiw cmanifrfW cftpxypojtionccaoaeflemult^Uc^fuperparticuljrt^qucfimfeoucifurefttawm^tumalnfetm parte bio et pare in ferfeo atn fc tin parteni bia rtumeru p:aouat: «para (ti(cp:oouat part^. igif quod p:oMiciturd}nummia et pare, crititac^piopoxnccompofIta <tiulttpl«ruperparticulan9. 'iPotTcsttempxobarertqutnquagcfmiSnonam^iuud. C-Sfoiilnpliccm rupct-parricularcm inquircrequebio fumpta mulripUccm fuperpar nentem componat. qutUbet impar aet bpxocittiue par teiue medietaefit e.forno molripllcf fuperparticularem cuiu8miiIitplict6tiunieru6fitcerpareoiaaaba.c|Ucp2opci:noi‘i(daDe:Dico ptopoitlonem multi plicem rupcrpariicularc oa» ebto fumpta tonflituere nmlnplitc fuperp8mmc.nam hl?«lnfe do» catur fiuenut quarzatue cct partee fimilce fumpte fm btparea t^uadiato aDt«a. lUe autepartee vnumconriituOmvnafupfluucailla.nqtiilcdueparrce pcrduoprcima^muavninoflcdftoant ptoouttum erit numerue ct partee.quare ptopozno coni polita aumultiple^ ruperpartii8.t inptc» pofito elTer quincupla fuperquadupartiene nonae. «7 C^fouUipUccm rupcrpaiticntcm perferutaritque Duplicata multiplicem fuperpartis Ollarem componar. CSuinoper fepageftmatcrtiam l?uiue fuperpartien tc que duplicata mulriplicS fuperpartinilarl coponat ct numer*’ dcnenunaepartee fit atnumerae fu b et vno maioz qafu ezatep cducatur in a: ct pzoducro addatur b. fittp torue o mantfefllj ptepoztzonc dao artTe multtplicefupcrpamcnc^cut* nuinertie denominae multipltcie cft tet dcnominae pariifi cfl atnumerue earu eft b:qu3 dupUeata att pico ciTe muuzpUc^ ruperparticulare.nant denominatio ipfiue in fe ducta ptoducet numeruetpane b< quod vt iR pzeccdcnribue videre ptomptum efl. 6S c^alSuItiplicem fuperpamentem: eje qua gemmata multiplexfupcrpartlcno ptouc: niatrDcmumrcftatinfpicerc. CAuerarnrperferagefimareotndambuiuefuppartiene quegeminata multipUcfrupcrpauientf conflituatzqua inuenta vt in ptecedente piocedenoficil. ClDatam piopotnoncm ui piopotnonco tquarum quotlibet fmtrupcrpaiticuUrcd equalca rcoigcrc. C^tnrminimi termini date ^oztloniea «b: Cita cnumerue fm quem fltm^ruBr TuperparticB lareeequake.deinoemaiotcdDcerurtnaetinbf banidetezitatatnrtivttnrrrdf eUnrplareeRu f—— nicnq^vnitarceincrt^ij^niuepoftdqulmtmarimuecotfifirf, maniftftfieil4»poaionffaooe(re p,j. cg pzopotrionem ruprrpartteulari tminima omnium que continuefunt mter numeroe ab drfqi ad et c- 5 qz denrminaraclt amino:eparre.connnuo pzopottionS fao dvno pluree §> fu numerue ctftnK^ gpgjb—~i l: Irnzcritcpptopomo bad mcompofita et 4.poztienkbU0 fad d fomptie fecundO numerum c.t qma operdccinianonamtcrt0eftaliquotic6inbfuctotic8innmaz)ifeftacflbadne€»rdade.qu3rcvt aadb.conflatcrgopzopottioneaadbcireoimfaminjjpomoncequarfialiquotfumpteeruntequa a b i» i™ leefueparticularcefccuncBnumcrumalTignatfizquodellpzopoficfi. 8t? uin» >0 CiPiopoITtio quotiibct cqualibue ternunis ct lUio tonoc in oiome euplog :atep cp il!i0mploo procreare; at(p cp puuo itiuctio toUDc mpropima mtilripliaratc counuc ^portionales proDuccrc. itemep cp illie rimilco cUccrc fupCTpamcularco. 1Cfigura cqualitatio quotcuntf^ terminorum cqualium. faaOTllL-bitaio 14.1 4-41 Itcqualttasi 4I 41 41 i^qnalita:' mII il II i: 11 iliee.l II tl II il il <1 1 ^pl. Bl II Zi 41 81 t<S\ 3ll !1!^ 4i 8i8l 5J.I 84 —Cl i! 3I9i 17I81 It4Jl !Il 5!9’ 171 8rl 145( 719 QuaozupU il 4l ts\ <S4.\ z5tfliot4! 1II 4Iitfl 841 l58l ioi4l409tf Sdultcrl. 1£I5t! 481 7ll 108 Iitfil i43f 1841 9<l i44l 118! ?l4i 488'719 Scf^ftii^ f114-3 3Z4; 43i \J>76\ 7‘58lioi4i £liot4.(K8o|i<oo(iooolx5oo| 5n5l liwl97if iioiliTxSl Z5o4'307zl40si6 l4^0(Cl5Uo\54oo:8oooliooooiii5oo, i5ii5 11?1"R 1jrT*iD H11'R 1XO \B\& Cfj^cccpti runtfKc. I 1|^JlmulTt fccunou. ;'02imurcdjb(0:fttu •f 1(t uratr1necqua(tatl3i i1'Brgi '|0 r'|2> 1 1 e; *. i£iuta <: i| i! 1;qa»i| i| i! tl li-i Jl 5i 51 tas DupU. jB.rrTt+r il\ ^8i 1Jl 6i III Xnpli. ICI i! 5: pi tl «1 tSl L5' CtnaciupU 1©1I1+• Itf! 1“T «nir 151 11' 4-8' B^plic5oetft* !£:411«1 4.) tl iil 61 J’ liS 8: 4 SciquaUcrl. 't' A' J81 lil i8i 1iil i8i I7l Cripiic5oer{\. 69iI'i8f tfl il 1171 Pl 3! JSefWirerrq. b9\ tl 16’ ii8.i4l »ll iX7. 5.6,48! 04 64;' itf. 4 6ua»»pUciuerfi. “T? 16 J. rJli 8: il ^efquiquarii. 5i;4<>l 5oi USI ^01 751 ZP|«I iZP 1TrtT z5TTR~i I5~ir Cl^yJbagotlfi/iDasi-^lpirtrcl fiCCTtiotfflaf£i«<j!»papne»crrtiDflasunt©iumi68&mir3turcflua- luaie pariicr imnJtate in fute arcbanie in imagine numere? veneritur taciti :ran?- rerum emmfi ptictpU.-a quiboe e£e rerfi picflutr^pemoiijpoiticnfi nepua: olfi? acmirabilie barmonia er in que taaoe reuoeataeunt oia-lp-iepter ergo mTftena prtfco? qtf auto: gencraliua inteoitipJtmfi in trina eemefiraeum ceqHaI»8tc.ae qtiquicf nte peragensfi mafarta funt vbica vete? pcepta aDmo no6 taotara Seucrmo.'t'nmQ eltl^timfi.SaSm enitbrtmfi/fctfm.Ieitifi dli^biimfi-fcCm bie/ terriQ Sintcrsoaoptre6tcnnmieqnalc8qmce<9vtmona?6-cpaoa;friaoaauttuiufca«alttriu8n0eri »lea.Btoipam« freepto caplccp pnntivnttatf qoS ceUoco fub ^ma Ofi? b.nelnMvtero ft6o fccpto capto ptunat fceamfiroQUquae colloco fubftaa-fmtqjra emeoo tertio^ptoicapio ptima zfetfat bifl«t tertia femeliquaa coUoeo lub tertia fint<p n.wco bmnpitimcep eqnaltfatc natos eupIos.TRj cfiaptimawitas ibftnt loeiei pilmactfctlaTnitasfinc^uplfiaopjlinavnitatf aeterna tfecunoa wiitas fuit miergo nfierus mcuplus eao b:cr cfi fetf amitas bis ffipts equef m«ftrr ptittia itertia fimulequcntur minfit feda miras bis er pma iterr-a fcmcl.crgo n euplusefl a» m.(unt i«tf bmn ^tmocoupU ab equalitatc fc5 quearmo-fi tictfi cft ^jpimc gcntti.Ctfbotcs ioi mofh-areDrtcefima et remfecimbi.Yu qia ao oirr oao p.crgo priceriml fccunoi cet pao atort oao a.et q: oap as n eto:ficutoaOJ.frgoprcrt5nufocrecuiiriopctao:aoatoficutorta80a.atopctaoftniulfuiitn et a 1 0 funt met 8cfl lofb.igif bmncotmucjipoiitonaturrrec mao bcogrm«cft oupUis igiF ct na» moupIas^tinoibusrequteibuebicargumttancimoouscefficapJEtqiuniptrmacapicftcrmfon ^ualiMSicarfmleSt argumeta. CSctSo cico «bmncuplis cottnuc^crcan rnplcs.copio cqualc bacolloco fub bmterrta Unca rtte? c.ccicecapio bmnftcolloco fub mfin tq; mtertie linee nirfum c«>io bmbie et ntcolloco fub mfitufuo mcco n. oleo cmntertie linee ce mples. Tfla per ricefima Wlacmuatefettaeiufolcmnfunrcotinue^potrlortalesifeortqzccquaturbTmfuDiwscuousI ac btigiF bct mfuctus funt mplus ac b.fcc bamfuperinunt mIferiusitgiF infertus ni ao ctriol/ qoarenaci^metamplus.funtigifctnncStinuempli.etelfc^pjcc^tistcemoftrationibVcc tnpUs nafcemr quacaipluep qnaotuplis quIcuoU.t ey cofequfnbus l?oc pacto ronlripliccsofs qut l^c?g)|:inio mincte. tCcnio oico ejr muUtpbcib’ coimuc nafet fucparticulares. vrej cuplicib» fd-qu8lterietrtpli8fefqtmfl^quao:upl(efef4qt»arti.iitapfcqufre0lbBpanicuUre8pcroVw e?natiiralif?fitlbu8mulfiplicfbMCoucnoouplo84repertifunt:ITnt«pconuerfienin;crjfffiraon! cuplus et mao nouplusiquaremoimtoifi c.£aplo equalee q« ntftcolloco fub c.ccmcecaptn m Bmul et ci^foco infenuo fu^itot®miferius-rurfuscapio embis tnfimulict colloco inferi4b « flnt^mce inferius n-p vicefima fecfici aciuuarcfeita flufC5 fmncotinac .ppottlonltur.fec cfi e^f ^etur et mfltcimloium eterg^m fimulcetinet totum ea cimtcifi.qu8recmao epcroiffmitionf fub temc.cefw Int^igfco tertia quarti et quinti cqualitare cotlnue fumi fm nume? tenfi rrmS imtUigo rurfnm mnttijiulitatj. tmnte: k, ftptimS othuO «non» BaSnaiiloSTmfSo mmnttim trina equalitatc mtcUijo oiiua quilibet tctminne intcllisif reninoo m“c?» lltisulo quojioco rubtet nOTftia.et II mplatet iq InfintiS cttenol libet, eoto toimnecouS.1? eqplM.io refuitiptla bemoltrationibua quotieotntnt tefumpte tqualitat» MqltaM-fientiq!? riotafanninieretnonnrablq.StcupiemplooeoocratetineSoconartfturPeMSte retuatibu (lletiii tertiia quibqf® loetq totunniiti taplanf tcttuiiii triplo:» q"Ibo q„“ nar ,Vrtii cunturcj: trina cqu&UtitctermiRot&fuinptotunifecunoii numerfittrtflcuiBfcpetrrtnil triplontin» tqttaoaipU cofmiiUrcr atriplis acceptis connue tribus tnplis fecuni^ eciacnttj ei^oaluarts rei^ cptrctni inpiotu ct l?ac analoijia mconrequeribus multipUcibus. £t rcfquattrrLldquumafef^qai; quarti Tconrcqu^res naiccnturcouerfisouplis tripUSiquaSiupUsetcsnfcqocnbusfumptisftmiU obreruantiabupUeconuerfis tnplfsquaPiupUs et reliquis quoncsopounatntu rqaaUratetfeb quopmultipltciu/eoniicrfotfiptlinumatqiniartmfiejitrcmopcqucmr nn0uloqu(Kp temo fupp arti culariu ctvbiipvt in pteteoecibus cil ibcm bcnionilranoi mobus. 7! C©3ti0 fupcrparticubnbuo vcl mulnpUabuo fuBparticoUribus muUipUcwfupcr particularco ct fuperparttcntcoroaaovcro rupcrparticttttbus aut muUipUabus fup^ parncntibue ruperparneres ct muliiplicco Tupcrpartictco pwcrcarc. C^Uitab cruQparticularcotcrmmict fumatur atcmeUilteptimue numerus ctafcmcltblmel ct flt ffecubus numerus ttta rccunofi trme piofcctionis ptcccpta nui ptecebfti factii eft etawcniat befquiwifiptfccPcticruntcontmucjjpoKionalcs.atpCTbccimSoetauambuiasptopomo esb» enmuItiplcrrupparticutartougtFetfabc.erantigiturcdrinucbefmultiplicesfu^arttcuiaresxr eooF mobo cf multipltcib^ fupparticulanb^nafcctur multiplucs fuBparticuIares.Si vero eSuerfw terminis ab cincipias per canoe becimaoctauSl^uius ptoucn ter cotinue ruperp3rti/tes.ricqs conilat ptimQ. Scciibum verS cobc mobo patebit per beciminonam bnius bats oivfneraperpirttaiuSi aut miUtipUci&rupparticntiii.itamria^gregatioamuiotiincepeafumtperbtctabccuninonmnpnMic: tuent multiplices fucpartientes.ila maioit terminoiptouenict fucpsrtieteoifici^ conflat j^pointunw suab^ vltlmis coclurioib^cDUt0dpotea.£i; e^lirare^icime nata buplo0.£c oaplis n1plos.£p mpl{0 qo:uplos:?bocpacto bcreliiismulripIiab^^fcquercr.eecKmultipltab^conuerflsnarctfuH particuUresiTterbupUsfcrquaUeros.fetnplisrerqmcertiosiet itabecofequctibusrucpamcula ribU6cj:fcquctibiiomultiplicib«~eteEfuEPsrticularib'’multiplicc8ruBparticularcs:vicifcfqDaU teris Puploe refqualtcros-fi tfefcltcrtUs buplos feftitcniostef bfic In mobA bc reliqui8.£ymuliipli cibus tuppar! iculanb^ittbe mul^lices rupparticularcstfcb ev bupUs fupparticuUrlbus tiipliruB partltularcs.fit cc triplis rupparttcularibv quabujpUfuepartlcuUres. £% fapparticuUnbvet mul tiplicibus fiiBparticularib'’ couerfls ruppartietestvt cc fefqualrcrio fupbiparticntes. £rfefiitertiis ftiEtrIpartles.£jc ouplis fcfqualreris 7etoupUs fcfiifcrfiis coiTIVfuBbipartictesct fuBtripsrriftes £f tuBpartietibus multiplices ruBpaittetcsivcceruBbiparciFtibas bupLas fuBbtpartietes. £f TuB’’ tTlpartiltibusbuplas fuBtripartietes.£>: mulripUcib* fupparttFrfbusiittbFmDltiplices fuBPame* tcsifeb er bupUs triplis aut quabtupUs fucparti^usitripU qusbtupU sut quicupli tuBpartites* £rcouerlteraep*rtteiibaBauemaltiplidb^aBpart(itibuamaKOturfuBpartiiteaXtertTibu0fite ceptis bec vbiqj lactflime patet.teepte(rs^cunqtvoles longa ferierfBBpantci^srtficdeunieppot tioiiBltuflcStiauemspceptatribustmbustermfefertciaptaooaiceoismafcFmrnmlrtplfawfaB» particulares tcb folli per tres et btres corinnc /ppotUottala^ecoipacro conuerfo orbinenaTc^uf fuBparnCres feb foUt in tribus ttribus terminis c5tinue j>po:t<omles>at bellis hactenus. 409SuBParncularw Aloic ZPulttpluesTuB ;b.c'pamculates 7t C0ataorupcrparncdarcomniulnplicc0atc^cafoantnmtotcsmulapUcceDoncc aD cqualiratcm pcruciuantircouccrc. aiabuertere licet ^pottiones in (Uas tifirefoluier 4busp:ocre3f.Cluo fiw inolbus (bcrcfol uenblmobus apBeat:in4buBibeeratmobU64>creabi.5mtabcbatt tenniptmuefuBpamcularcs ct fit bcquaUsaTpmuenfieniBretabbfoUafoTremancatefbcctolUHJrboebist remaneat f. oicoDefefrecdrlnue^potrfonales etmulriptice6.q’nntc5rinncppo:tionalesbeclsraf.eap(o cnig bIt tres terminos tnedtinua ^pottiones ab e: jer ilHs fuperioti aggregabimobo jenteteruntec {icebffibuscorintie^tpojiionalcsctfintlmn.ctqilabgficuiaabbtmabbflcutbabe.ittaoftter tgtrurbeFf*unrcotinue^po:r(onales.crq:babaBbFP°^^itieiiruBpamcularisBb«citn3recram buiuscrfceabFflmilisbeftoiattonismiiltiplec. qnarcrnBParticuUresbaccifuntfnruasmftfpliccs rcbucrc.fi itaipfupcrpartlcularesoatcerantfefquftmierefoluicfuntintnplasfifefquaUere inbu* pias q’rircbucterint in triplas eobimobo quoptlus apiauanterepruagentnabniusptobabisillas multiplices rebuci in bupias ct ouplas in equalitatFiquoo dlptopbfitum. C-dfoultipliccs riipcrparriculflrcs mfupcrparricularcsit muUiplicce fupcrpsrtictcs '^in ruperpartietttes rcfolucrc. itSiUi abctres fminf muUiplices fUBpartlculafes-ct bcqualis a4 fuWVrabafab b1rdinquatur cteomoquoinptcccbStirubfirabatur.fibabacritmulfipletfuBparticularisvtriqjabeobFiificrc» benoiacus Bquaotagefimapttmibuius factC erit ^ofitfi. et itabefuBParrifttbus. St il muUiplcr ntmaibtis senoiationis no fiet boc^niarefolutioncfeorefoliiein^ptimolR multiplices vnoBiinorf oenoiatiottfsbenoiartonepartlsfcmpereaslmanentetbonecabbocvcniamusvtcablfitbcso^tio multiplidsctpartis ficq^quob^onlturfiicUeapparebir. C^uallibct fuperpartiented in fupcrp^cubrcdrcDact cft poffU^lc. 74 CiRam qt per fepwagefimiptlmam buins fucpartfcntes ejintBparricutarlbas naTditor. Ibco fine abcbulufmonifuperpartfifes-toeqtteturatfubftrabatun^babbCtreUnquafeTbecebts abetet relinquatur fnvenient per (Uam reptuagefimamptitnamruperparticoUrcsa quibus nafcebantur. at non oportet tbremper in ptima rebuetfone contingere qnob tamen nunceStingebat. feb altquabo rcfoIuFmrptimarefolutione fnmnlrtpliccsfnperparttcularcsqneperpeebente fnfuEparnculares tanoercbucenFiquo facto faafi eritptopofttfi.Silqli (ftfttcparti&es aiitimultipUccs fuBParticntcs C4 1mn 5bb bef abc P«4 tt intiltt()l<i:rKetKHtlcn« per ^ceMntirtODCtC infucparticnt^ que tep reoacerur in rueitlirtictdari TtfiDctracwowbrcntaneatcqnecontlneMoetetueparteswel iufucparttoiUri neoe«tractore mancateqoccontineatoetciU6p«rt<.atftrema»eat equi contineat Dejparte«.qj illa perracrtotfi innmciineHnon pdmfeme itirupparn^tes refotoamr aliqii occurret rupparticularta.facrttt appa rtbitpu3foru&.0aarmoico umiruperparttoilartect no multiplepqt ej: multipljclbua non(me otatcfacpanientee Tee fuperparticalares per reptua^eUma l?ulua ^i^fitur. ClDatad p^opozticmca quarcmtip 9D cquiditatcm rcDucerc. CfUmperfepniagefanatmtambmoamulrtpUecefuppartlentcB rcbucCmrlnrupcrpartiftce et itntlttpUceafoitpantcularea in ruperparticuUreBlperpceoenrffugpamftco in fuEparticulares. et per reptuagefunaTecmioam buianfuEparticularte reoucStur inmolriplices ct mltiplicea in alte» miithpUcea oonec tn cqualttaterdhtute^rint.omnce teititr tn equalltate reoucfitur. iRoni clem&opBrt^meticee ^otoani Unie. fioiciao cft conncmo cjircmojutbabiniDHic viiiurtp ao mcoiu. 4&coics tatum qucoam futit pzmcipalcs: ci qucoam pzinapaUbus coUaficralco, •(>:mapalc0funt tres, 'ftzima cft quaoo maioas/mesq/ac minotis cqua« lc0funjoifFCTctic.ctbccDiaturSn'tbmcnc9nict)icia0.SccuDacftqu3oo mapmi ao mctifi5'caoc cft ptopoJtiotq fticoq ao mmimu.ct bcc geometnea vocatur Cema cft qn mapmiao mmimu caofi dt 4>po:rio: que otte majn mi ao mcoi0 ao oif ferenua mcoij ao mmiraij.ct bcc mufica nucupatur. Collaterales funt oao.-^^ma cftqamapmusao mcoiu ficutona mmotfi ao oiffirrcnria maio:um. Seda cft qfi mc oiuaaominimutncutoua minom ao oiffcrciiammaiotiJ.Ccma eft qn tnapitius ao mai0 (tcut oria cptrcmoru ao oifferenna maioip. Quartaeft qu mcoius ao miniitiu Heutoria ejetremoru ao oifferenam maiom. ^aumta cfl qfi mcoius ao mfmmu: iTcut oifficr^a ettremoyao ojffifrcntij ffiirio2um.0ctta dl qfi ma]cimusao mimmutfioit oifltrcoarniftoruao oriam maioju.0epnma dlqjl marimusaominimu: ficutoria cwrcmoju ao oiffcrcnna niaiojum. iOctaua eft quanoo matimus ao mimmui ITcut oiffcrcnria crtrcniomm ao oiffcrcnriam minomm. ggj, gg C0ifucnnttrcsnummmambmciicamcoictatcoifpofiti:criimaio:isaomcomm 1—7—7-T—fmmo:p:opo:iio ^mcoii ao mmimum. CSmtafacrresnumminariil?incticflmeoicf3feoifpofit{quo9m9>rimu8fita:olcoaa»bminozg Vjygyeneavotnoneg-baoc.napcrolfflnftioncaaobetbaoccqiialeafuntoiIcqtlgiturftTbruntvfcos pofWepbetcipfte aooltis numcriscqualtbus fcj olfferentiis eojfi.ergo per oeclinSfcttainfccunoi iPpouto aao bmtnotcrtrptopouionebao ctquoe eft ptopoftnim. iQ C^n^ocmcoictatcouocjWcmicoiunctinnrcoountt^tummcomsDuplicattjs. ^ ^^’C*«mEOilfinltion«pUtniat>mc»lfietmeoiJa»ejtremOeqaaleaftifttotre:f«lturBfccuno3p:iin< mcoiuseft eprcmotfi cemficto? meoietas. oupUeat* Igitur cquaf ejtrcmtsplficrietqti eft ^Jpofitfi. 4——7 b* Clln caoc meoietate quoo fub ejctrcmts continetur cum quaotato oiffererie cqualc eft quaotato mcoii. c1CAAmperptcmilTamOuoettrcmininulfunttan^oapluBmeea. ^onaFergobuplusmetHiq ftt oolnifuspouoequaliaquop mcoiuseft vnfit'! per ouoinequalia 4rintaccTtremitetEOeclmanoni ptfmi erit quaozacus bonlus eqaaUnm qtum quoo fit cv atn ccum eo quoo fit ee oiffereria aao bin oiferentUbaoc.atqtoiireretitieiUerunteQuale8:tanmmergoeritquaotarfmeoif b^mquoo cStinour Ibb cetremis atccumquaotato Otie meoi) ao ettrrmfi: quoo eft ptopofttft. C9ncaocm quo(^ meoicmtc quaotata qctremorum :ouplum funt quaotato mcoiT ctquaotato oiffiercntic. .g C®ntabcnumeriln^m<olewteoifpofitirfltqjoquaojata8ctc.oicooonplBe(rcquaojntome fPffequaotatootleftinal.Sttenifqtffitotatnebis-tqtjaOtarabmcoflbfsfintetqtentpcrrccunoJ boiuoaetccoiunctitaittfifuntStumoaploeb.etperoedmSquintlpiimtoifnmuttiftfuntq-tum ,quaoatfitotiusacfeOTpercanoFoedmSquintaceritmcPietasotffimul.igiFperfecunoapartl reciiocptlmi eao 0rao fequalesfont Oile.lginirp«r olffinirionFebacmcoicfarc mcoius cftftiter 0 etf.tqtpcr^cocnteqtfftteramcfemclcftquaourootierantijcft^tumbinre.fgtFeaooitruperf ouplfi quaotatiotietquarcto tantumaooctruEc.quareTOupIliquaotartoiffcrerfebao'aequaFo. IgltnrquaDtatuebcjefttneDicraseciiquaotaiooitrerFncbaDaeftineOietasotqOeftptoporidi. 5effrium numero?in bac meoictatcoirpoffro? ouob^ettremis oatistmeoiu rmre. CStntabouo ettremiinbacp:opo:tfone:illcsftmuUot>oat(p aggregati fumo meoietate que per fecunoam I^uius erit tneotu8petitu8:factumcp eritptopoilrQ. J t X numcroy ao ofiam: fumitur trcsnucn' ifi caocm mcoictjrc.cnrmcouillo^:aooiffcmpm:qucinrcrfjtrfmoo paHiffnaro? ^pomo. mmo^rrr^.!.^ ft «8» b“I Piffcrcntta bac c.fumo item tn caMmmrotetatenumeroa f« b f- ^fmarirnua-eicogao tufferftiamf abaq ntUam JWi ti of^f cnl otfferctfam fiao bqfit k:cutL8 mesietae fit l.ctco^ifituB e^^ ^ ^efunt a.igimr bfertam fetun&la ao efleut mao b. r ,M?m‘?Ji^'^^*“^^®*’*’**^^***?*"Si>Eniutjntnb 89 ffom8aorort;ncutea»bcctra<fueabt>ctra a^K rergo eao evt bao It. ergo eviceflmaptimam fccfioi tequS jjpottioaluatati m oimioum oitrcrcn tiam fteut torum mao Urotam oifferenti8m.quoo eft piopoflttim. 6Si fucmii treo nucrim bac incoictate Difpofio: OJrcmo?!/ 1 mcofj ao oiffcrctiara caiiocmcircpjopo!iioncmirapoffibiIccft4n3io!cm)}onucmrao!cm:pombilc. CSintircsnumcri«bcml;»cmtOimre5(l(mtl:ctiimartaue;nolirn«ntiaaaoblttblaO(ni». Oleo uiipciTibae eiie caocm ptopoztiocm aao eibao o.nam fi iopoffibft fotctrponsf ergowfUa ao ctanqrpaDo.crgoEvicefiniaqumiainfecfioiqi}fitci:aino:equacftctqutfitejEbinc.8tDOedmain quarfajiip;imiqi5htcpainotmcft$rumqbfitejromfcctoi(tbqg{finretf.ff0percanoembfnc tincltq-mmqDfttcpcmDicmfequtfmigTb^equabaturergoooinulcemeffimBletgbfimBLfltf clhmpomblc.qoficoftcDo.»amBfcpttmamfccfioif80cf[cutbaoo.ctpcreaiio«nb80gncHtcao o.fcocnonamlccuoi^jpotttoboeomaiotdl^^pottiocaoo.tgift^oaiofaocmaiotentappot tioneb ao g.quarefifcqiicfbBcafcmecritmtuotg.quarectfnoncquabflturgttj.etftfatmfo» bcr0OEOccimarcptimafccuuotccttacrttminoig.quarecompontu8Cj:cifouobu8 mmoitbuanoti cquabinircopofirocjrgab-afftfponifmaiotb/weefnpofitusetfcquenircopofitogtbicrgopcr oeclmaiionaiufc?5i^)po:rtofaoeminotcritv4>po2ttobaDg:aii^oppofitummo(h'atumeft. coftat ergoqc.oictUeft.CScoejtrcmiaoc;:tr<rnmm^)po:tiomaio2polTtbai8eft§’meoiiaooilferentiam (Ubv>mtatctrc6?tmucponan«terminiquicrutmbacmcotctatc.ctq:ewremifcjtcmar« ao tnfta temeit^po«iotnpU:feomeoi|binarwaDvnitatem(qucotffirctUcft)flpoztiooupla:?ftatcjctremi ao eptremum maiotcmelTeptopotttocm S” meotf aoot(fcr&um>et ttaupoftTnitatem in bacmeole tate num^i apte oifeotinue fumatur.lioiribtU» eft «ia extremi aoertreraumjfpojoo mio; ffmeoif ao omcrctiam.uam acceptio quiburcutbtribuatermlnia contimiepoilvmtatemzmaiot erti ^poitio ineoij ao rintarem queetit ^tSerentU^ejxremi ao eytremum quo^ oi/ferenoa femfi dl^btnariua. CS^rcsnumcroembacmcoictarcwucnirequop^poztiofitsmcoy sDOiffcr&iaffl: av> pzopo^aonem cptrcmoaim Difib-cniia fu quislibct Data piopottionemaioi* CSintatbnucrioatc^potrioietbmaicMtaooobaDatlutccaiettaaooobetSatOananifcUum eitocaclTetrcenumeroa mbac meoletaiecoilitutoacfibdtcoteoifFeretUoaocf caoaupioi^oico oiiFcrctiain.PPo:t(cnia cao bmco0ao oufcrctiamiao atpottionem oaoa efle oata jipottioite bao a maio:em.nam^poztiocaoaaooitruEp:opo2tioncmcaobp»>pottionembaoa.ct ptopoztiooaoa cTt mai02 jipozncnc cao a.aooet tgtf ,|>po2tio oao afuE ptopoztioemc aobmaiue pzopoztloe oata baoa.eflirai:poiiTerctia^o2tiotacaDbaop2opo2ticiemoaoaoarap2op«moncmaiot. e0ic^n^cm^ cytrcmi DupluoJcrit mcDiue Diffcrcnricmplu0:ctc6ucrnmr.q»fi3lfcr alrcnue maiu6 ouplo fucrittcncmcoius ttunuo mplo Diffa<mae:c| ecduerfo* CI'lHamfIflmatimuaritoupluabm[mi:critboupludOifferenria9,qft3rcccpbfoifcr^tdiUaconi tutus elocinoifFereticcrittriplue.etriccfttripluaaooifiercnttamcfipilituat et: Illa oiffrretiaetb: «rgobcritouplueaool/fercnam.quarcapftitumaejrbiroupla oiffer^tiaquccquatorbitepinuaii eturouplu8aob.quoeeftptimum.ctflacfttTiplu8aob:ioembeqaabifviuoifferenfia?.atbtvna equaltum oilTerenttap ^Ihtuut c.lg(tur couplua eft aooiiferentiam.et (i couplua eil ao oiffer^azn erit rurfus atriplus ao b.nam fl cfit ouplue ao oiffercntiacafit pfHtm^cyciffcrftia «btoilferentia illa cquablrur b.feo bcum cqualt oifferetia bis funipta pflituit a.tgtmr atriplus cft ao b. et vniuer: faliter^ marirtii ao mimumptopoztio matot 9oupla inueniatur: mr femp mimus ao oifferentiam minoi$ouplu8.igif •tmeoiuBcjzmimoiojfferentiacofhmtus aooiffcr&iammlnotmt 9>triplus £t couertitur vt inalQs quoc^ monilratum eil. a CBansquibufltbct Duabus ^potnoibueifmvtrac$ in Bntbmctica meDietatc tres i termmos o:Dinarc:quo:p sDinuicem oifficrcntic fint cquali». C;Siutmtnimttermitiiouarfloar3rtt 4)pozrtonumabtco.etoiffcrct<ctIlobfintctf.fumoite?g nucrura aliquem quem epiter nomerentbifferentiee tf«ct fint numeri fm quos ipm numentbet I;. ocinoemiiltipIicoatbpCTb^f^eni3tlfmtetmultipUcocetoperlict.pucnl3tnrto. Efeptimam ^ fccfioilaemvtaaob.etnaoovtcaoo.ctoifferftialaomeftg.tfiHteroiffer^tianaooeftg.fitgiF e geftpanfumomcoium inter Ictm qui coilabit e^metoimioio g.etmeoi6inter net oquicoftabitei; 0etoimioiog.ctcriitrrcstermimlTmcfiruomeoio^/ppoztioemasob/et tres alif neto conQcra b ro eoi^meolo fm^ipoztioemcaoe in arithmetica mepietotequo^aoinuicemoiiferetiefuntequaUs j9a9 D7ts b——<5 c4 8$bs bc4 i t5rfnctnrfobtnecwrp3rutfrimo:boctn8uWMtufllpcrrcptimfltiiffcljptquoileemflrfmt flbniTtnfi fftbabitotooopla/nlioquinpereanomnontcneatfto pcrfantcmfcnipcrquf^-ppatomarimjao lutjTtiiin da CTit dius quoc cotinctur fub niayittio inieoio ac 18 <ii5 cewincturfub racoip xnttnio/ itno iroimumrcieratl rdmunedK CScpitmetcrmmo0irmcftigorc. « Cbeccp gfommica .Sini abcrrce numeri in geometrica mcbictatc ab amapfmo co?. rubftrabo oiiferencianimltictum^Jiltrciitniump.erpatebttTt piiusque^j^ttioaatbmatimiapmlmutcam drepiffcrentletptrcmo^aPClfffTcntiammaionjm.quarepcreiffnirionemconRabirptppofitum. ClSttauamCfitiumrcltatcrtilanarc. , , .,* CbccettameEgcemcirica.Siuti^iuaPaftnumcTlaJbiclngcomctrlcamcblcratctctmitipiScifferentia qpcatur mcoio:fiarq! ccinpefuuab.oico a/p/bciTc in pacvitta mecietatCfnam quc^potno a«e bmaptmi a? mimt:m:ca cflaPniuSicvIccfima butue eilfmtia aap bCTtrenioJu ao Pifferctiam PbminotiimamcquaturPifFmnrrcbaor.iguurpcr oiffinitieticmfl/ofbfuntlnpacmcPfetatc.nctSi conflant octo colla tcralca meMctatoa tnoiiDratc* CCnufnnumcroJunimayiniiaonicoiumiflti^ ciffercnticmarimi aDmimmumao <so Differentiam mccM aD ipmicanDcm cITc p:opojtioncm impolTibile cft. esittt a(b,' ctres otuerfi numeri quicat^tque:;; afit mapimua/ bmePlus/ et c mimus: et tifferentia aaPcntP.ctPiffercmiabaDcnte.DicolmpombilcfnecanPcm ^poitfoncaaPbmaEimiao mePifi qiieoapePlircrcntiemaEtml^t minimi aoptffcrentiammecfl aPmimum.tiflmfiea^jpctttoriraao bqueoaPetotinsfcjaPionjmquePctracilaPPetracffiteritcrcfiPuiaPcrcuPnCtqucaaPbtonus aDtorum.eftefiifncvtriuitprcflPuinii.quarcattbircin funtnumemsatcpequales.quoc cIVeontra l?Tpotfjcflm. cSiintcrDuoanumcrocrumarurtrcamccyivnuafmaritbmcticammcoictarcmt m ct vnu0 fnt gconiciricani:cttcrRU0 fm batmonicatntaritbmciicum maictmu: xbar? monicum mTmumiotcp inrcr illc© gccmciriirm mcDiiim cfle ^ipoitioalir cft ncccffc. esit 8aP ciJpwrtP qoccBtp qucpotcHin Puo equa PiutPl.vt crcmpli caufa quaorupla inter cuius Wremaper triceflmanonam bulusfit inucntue bin barmonica mcPietatc Puplo o b ct ctjsueniant 8fe/fqutptintercr5cinca8cm mcPictate.et quiapcrftptimnnifccCttoaofvtaaoc.'craa8cponta (Apartibilts in cuae equas ^pottioncstparticturenam o08 ftn cquae.Pimoetur igitur per g<eric igitur eBPgVfgaPf.quaregperPiffinittontm nictius tflgecmctrKus tuter oif.ctquia ti»fur.t parcfltfprojfifimuliuncfetum capiemctictafemqucfirbtqucpcrquinrambuius cftinartibmctica mePtetate inter 8tf.babcmuelgif inter ©uos numeros oerftrcsmc8ic6l?gefc5binarir!,mctica mePletateigingcemetrita/teinbatmonicaPemoflraios.qSfupponcbat^pofitio.PicolgiturPeffie inaiimummePlotum:ctemtmfi:ctgnic8ium jjpojtioeliteratqsgecmctricuminter pte ncitiquta Ij dtmePiHs inf p « ffm arithmeticam mcPictate eefm barmonicamterge cquapragcflmaptiniay ljuiu6bmaioteftg'e.ctque4>po«loPaPbca<ft«a®f*i8‘i‘*'’pervt<c(lmaqulntarifctaoiqi5fitej:e inftcqufi cft fl qui fit eje bin c.fe© quia gcflmcpiusjpottioaUs inter pafttgitur pcrviccfimafertam cipfPfmqnoPlitei:Pfnf:equBeftclqulfitet:ginre.quarequetfitc>:glnfetihefl ^ttmqucoeEb lne.tgtturperre<6oamparrcmelufP<mvlcefimefcEtefeennplb5eftntptopoiti5alC8.quarebertt msE^ustg mePluspjopouionalistTemlnimus.quoPlntcnPebatpzopoIltio. C®uO0numeroe (juo? jpcjtici non fit cata ptcpostioncmaio: /(nucftigarcfintcr « quoo fm Irco oictao mcoiotalco mcoioo alTignarc fit poffibilc. C^btmptopoitionem quScunepPluiPamus per (eEagcflminonam noni In quotilbet gspottfOReet qu8pPueflnrequale6:qaeppofltenonnt8atap:opo2tiocmaiot.fmigiflU8m ;ppo2tiocm ^pofltam ^cromePtumbarmonicum ftncjrigcnriam tricefimenonebuiusctolnoeonnmilircr operato vr m ptlnclplopteceoenrle factum ellmueniam tresmctlos mooo qulin p:eceorntioicru6cll,lnterPuos numeros quo? ptopottlo nonerltPataptoponlonematoz.rici:pconfubttpiopoficum. C^intlelementonmt arithmetices ^ottani linis* I C?acobi faba Stapulcnlia Ocmcnta jfcuficaUa ao darillima viro 'HicoUum Dcboqncuillc inquUltoiium ifiicfioitcm. Eatucridanllimc vir nulli meas ei emento? muticaliu qualcfcnn® fum piiuooicarc vistUaoiqt ptobatao eognouiffc.^uio lO me mime lattrel/oemoHranonesan quibuovcl foliovio feicmiecofilHtmo pto bari non potmiTe.'® ern Ii piefennii inoicia parai ouao/reputaf(?:bl nbi et atiqwo filmopae muficoo comenoare laboico iurc valebunt. Jfocrcurius Camytas t^tolomcus iSjpbaio Jorfracnuo Cbcbanus (Suboliocs Ovibasorao Samius ^crpanocrlcfbius IbippoTus ampbion tbebanus Jtreaon Samius 3nfto)ccnu0 lUnue t>:opb!ailus -ficnotcs '|^bllolaus f>Ttt>ago»cus. Bnon Ici^ms eihacus Colophonius. arcbftas tarennnus jfov&aolibJTSiUB fCTmotbcusmilcilus Blbmus flTotebuslroms ‘Bicomacbus Biuus 0camn’ Cocciusbragnistib:?): -gelato -abarfias anftotclcs. l£C ntnutuni c^piunnii quo^ qco cuiTia lutmoaa u rcDDioii ifipcs mter quoe:i>uoopccpto:cs mcoe^acobu Jlabiiiiu i^acobu lurbc imu onumcro:tan^ ca arte pofteniati victuros. Comeoani zea mmfid ciuo cfficcr*. '0Ttbago:(a ent animo? ferocia nbiio/fiDiburc^ emolltcbant.£fclcpiaDC9 frcmciio vulgi feoinones crcbJO cantu compcfcuit. Joecp tuba furois meoebamr. ©amon pytbagoticus cbtioe etptomocpetulatce aoolcfcctcstgrauionb** mooUlis 00 tepcrantiamreoujat.fcbte Tvulncraimullca moDUlationcairawtann^tas Capc quoep fuauitarc fcbias/copcnoiciiqj doIo:c0cmeoauit: qiJ bifincnias ibeban* tciaflc mcmoianir.tCbcopbialhio ao animi pturbationce moocrioao muflcoe aobi buiflcmcmotafmoDulco. flccinmriaquiDc.dlcni mufica vtqucoaniooCTattonio IcK at® regula .sSuamobJc bono iurc eoo noebat oiogcnco muficoot qm cfl citbara ao barmonicoo cofcnruobabcrcnt tcinpcratatanimu gererent mcopofitu et ptoifuo barmonia vite oelhturu. i'cnocratc0 ozganids moDuUs Irmpbaticoslibcrauit. £alcs Cretenne fuauitate aibarcimoiboo/pcflilennac^ fiigamt. zerpaocr et anon aones ?Icfbioe cantu agrauilfimiomoibis leuatTc: omue Scuermus auto: cft. IberopbilusmcOicus egrotum venas muficis penficulabai numens. SCrmotbeuo aut muficus oumvoluit efferatu rcooioii al^nomao arntatp furente ati aliter eu libuit ab armis ao couiuia rcrratic emoUitu.Cbwciue^tpbcus fera? fenfus fioibus/catuc? infictit.boc cft fermos bolm motes leges ao citbara canaioo Qo moocrata bumanitatf rcoutit.flCcrui fifhilis capiuwr ftuoiocpmooulauonu octiJ ncnrur.fCygni brperbotei citborc caiibus allicmtur. Clepbatcs inoi otganicaoulcc Dinc permulcetur, «uiculc fidulis irretiOmr. teneros aobue infannu fenfus permo ueni catus/crepitaculaipvagientij fcolt.Belpbmos fuis fibi fiwbus coaliauitanon Serpaes eltibus rupuntur.Sepulto? man«cantibus etataiur.^n actiaeo mare citbara pcrfonaremonotatur.^egaris cubara pcrfonaifatutrt ao euti^ibct puirantis ictum fioiemat. £tpolTcm pletaep raUa vir danfftmc ao mufia tionc aoouccre.at tot/tanrift»? rcccnnu tptifco? eomcnMWauteutatibusubt nue ob meat® bonanT littera? fboiis oleata oifdplmatequo fufcipias animo tms auipi ens lucem babitura. fi£t me vnum inter tuos dietulos marum viriura/tui® nominis obfcruatotcmcfTccognofdto. ^ale. CTacobue Stapulcnfls jacebo JLobimo 15flCObo 1Curbclino4fiiulTci0:fuio cbanfllmid p:cccpco2ibu0. ^ODinter o:atet£atep rbctoiat lo inter cantozem tmuftcutn interefle vo lunrtnec^ ozatozem quem^ oiamcrm qui locm rbeto: non fit.ita vcflruj femper lUDicium fuittvi ne catuoz quioe Dici mercatur Tncptqui iocm mu^ flcu0non fuentipuUbtc mimoo/ct brftrionce acantozum bonefto cem fc qucftratc0 tan^ Cpicurcoe afobaa mcnfa/caltocp pbilofopbotu Dogmate, nec m* iuria nam t>omcru0 oiumue t^octa vbitp Doaum et feria mooulatu mtroDucit can* lotctn VI apuD *0DrlTcam vbi ^cnclopc -pbemifi floicine ao ncruoo verba moucs tem bia verbia allocutam effingit. ^tctcrca illacbzimano oiuum cft affata canentem Jbemi muUatcncebominummulccntia pcctuo facta bominum atep ocumtet que lauDi autotibuo aoDunt. £|c iio pange aliquiD. etqualca Ibomcruo piobat: voeminime talea dTc oubito vr qui ouofl avobio ptia muficce ruDimenia percepenm. eiuaptopter qd voa noffroe labotco c^raminancoo cdmitto.quo0eobbcnciuo me rufcepiirc fateo: :quo muflcaU fdcntia ncc^ opuo grc« cooncc? laonoo quiDc vn$ elemenna traoitam etTc Icgeri :atmtroDucnonco $iplu« rimae mter quae ea omnm nobililTima edquam Diuue 6cuennu0 36occiu0 fui mo^ numentu rcUqmt.qucm vnu in bac re pzcfcrti oclcgi mcozu IhiDiozu Duce .Bi ergo ptobaucritie fatie micbi clt.^n re enim nottra malumua aliom lUDicia fcquii^ pzo^ Annaui quiocmuflca fcripfcnini 5nltgnc0CK rcccntioub* qui DC caoc fcriprcrc. ©cmocritue Iblato albtnue Bjcraclioee ponticue arffiotdco ^iuueScuennue KbimotbeusmilcriuB Cbcopbtaihte 33artIiU0 l^bilolaus Wicbomanio tf^rlanue srcbitae tarentinue arillownua SugulUnus 0UO SCbeoDou ptalomcus ambrofiue SaiitbuoMbcnioifo iSelaftue. Argumentum quattuoz libtozii muficce* Ci^umuelibCTinteruaUamulTdomoDulationibue acdmoDaDircutit.muUipIet::Du plarc/triplarc/quaDmplarc.fuperpaniculare:fdquaUe¥/lefquircnifi/fefqmoctauu/ bie fcTquioctauil/tCT fefquioctauu/quatcr fef<5octauu/qiiin(ic0?fcicice fcfquioctauu eSecunoue dc tono/integro toniDimiDio/rcmiiomo minotc/femitonio maiozc/co mate/ fcbifmatc/ ate^ Diarcbirmate. cCcrtluoDc ferquitono/Ditono Diaieffaru/Diapcnte/oiapcntc ttono/Diapafon/Dia pafon ttrtremitomo/Diaparon et oitono/Diapafon et Diateflaro/Diapafon toiapete/ Diapafon Diapente ttono/oilbiaparon/ac integro toni -r cofonantiaru ofm oiiniDio. £t DC marimaram barmoniarum cenfonaniiieict quarunoa mcDietatum. C^u3riu0Ocmonocbo:oo/tctracbo2Do/pcniacbotDO/bcptacbo:DO/octocbO2Do/ pcntaDecacbo2Do/oiatom'd0/cb2omatid0/ cnarmoniciemcloDue. CtDcmcloDiaru mooie. et bcc fub bteuitate contracta :argumentii libti funto. i C5«obi fabii Siapulcnfii! eicmcntoiu muntalia 90 dariirtmum viram «icobum oc baqucuillc picTp Dcntcro panllcnfcmUbcr pjirouo. tqtcraalUim cft foni srauis ;aoittcs fpadoium babimoo. Spaaumvocamusncrau/cboaoam/cppiratamacrcm/ctqmo: quio llmilcc(l;a quo fonumcUcimue. atesultiplcp mtcmallu cRquonce mafua fpaau continet minuo plurico.Ti biotet ouplejc intmiaUum oianir.Tt terno: et oiatur iripleje-vt quarto et quaotuplej nuncupal.t boc pacto Dctccpo 5ntcruaUuro ruperpartieulare cll cum raaiuo connnei minus: et i^Scll que memur totum :tpfum aliquoties fumpta reftitucno. Bi roaiuo continet minuo et eiufoc oimioium: fcrqualtcru nominatur interuallura. Stmaiuo continet minuo et minotio partem tertiam :oteitur rerqmtcmum. Bi vero cotinct minus et cmfoc minoas partem oetauaicpojoonm; rcfquioctauui® nominatur.Et ita reliqua fupcrpamcularta iteru.iUa:pcrfaale cft oiffimrc: feo parfi mulleo conDUCuntptopolito. „ Buperparticno autem oiccretur mtcruallum ;fi maius connnact minuo et minoas partcsaUquof.quepartevnainnonefficiunt. iBurocri funt aolnmcemptiim ;quos fola metitur vniias. ©ifinitates. •Bisquio metitur alterum :metitur et omne menfuratii ab lUo. !•Compofitumin ea rcroluitur fimpliciatq: qmbus componitur, j(Cuiuftibct numat pars eft vnitas :ab eo oenommata. 4®nitas in quemcunep numerum oucta; ipfitra ptooucit. 5«auioquio metitur octractum et refiouum: metimr etotum. s^uecuncBVnictciocmcquaruntiTcaintcrfcruntcqiia. 7et que inter fe funt equaiciufocm funt eque multiplicia aut fitbmultiplicia. 8eoaimquoatmtotaruntcquaictoimiDiafiintcqua. _ 9ESuioqmo bis auctum aliqulo tranfccnoit :lO vitra ilbus oimioiu efie ncccflc elt. 10 gt quoo onplieatum nonimplet iittcgruroiio ptofecto non continet otmtotum. ‘I i0rone totum cll maiusftta pane. fntcmallum muWplc^binario multiplicetur: io quoo fit c% bac multiplicatione intcruallum multiple]ccu» QiOBplatione. ajiiUtplcj: tatertwUum. CyntenuUom bin^o rouIHpHfwceft [ntmialli OaWtuWnnn t*opIareque quopacro PuplSoa fit ouDumrmiapiopomione qumti Srltbmaticeanorafndrepotdl>vtfIabntintcruaUomquooc(i« quoPbuurto molrtpUcarc tubcamur :ipftun p<r terriam qtiliirt mulrtpKcabtmu® rt ©admue ain fe nbtnf«zou«ni«t<p oteridp pioouctaaitn ©«c©aples Inteniallum interuaUo pioeu«ml6 ab, num©ucimtKalnbpwumiat9pz©©octum amtperredaniqBai^qoep2opozdoDa&*- qo«©mo©um ptopotiitpzopofirt© qnoo moco paaloente©icto:binart© maltlpUc*tur :ntcabuplum ©cettapwpomooaotteg© calco intcroallaeaaeeiremuUipUy* "fiam cum aaacnwltiplerfit tntf^U^cmgeminarur mantfcfhim cftmuUfplejr aoat multtplW qoareper©tccflmifecun©afti noni:ann?«ncn«B ftantn^fiotutn efl inieroaUnm compofuum tnuIrtplQ: efre.? piopofite.Sc© io«n aitfCTpic ©emonnratur.fiuoniSenim que piopozHo aa© bea cft aa© eet eaa c. nam Ttrasozovou tionie wplareaa©bmceleta8.'|ftiopo:tioafitaa©bporitaeftmBltipI<tr.1gtniretpwpo«ioo8©e muUipiesefhmmfigimr Pcr»i/Fimtion^nmliipl(de;cjpfumo©eibi8wlter©ei©cmccp0.£tcoog quoqj larec rotiea i^rar ipfum e.quaret cper pzima communJ conceptfi etiam metitur ipfum a. Cultquia enim alterfi mctiturtmcrtnir et omne queo menfurate eft ab ille, eft itam per ©iffiniria«i Interuallum acmultlplcy. quoo erat oftenDcnoum. esifuennr fcrmim^pornonaliter conflituti: eum pjimus fuentvitimo coparattis np:imuDvInniumfucntmcnru0:meiJcnirctffcunDum. j*'|,4l 8|l i) il +l®bcire«termtnipiopo:tio* IbIcII0(eI, aI, nalea qaotfi anumeratb. multiplejc efliclcaa. toii^taomnr.mMnmplitu»irotainrnqulKc!SMi"o““SS5Sl^'' I 4- CSircrmino2umteruaUtp:imu6 aorecunDuccmpanniarcfcvttcrtiuo aoijuartii babucnt: quot p2opo2tionalitcr mcoij pumo ct fccunoo: totiocm temo tquarto itu tcrucnircncccfTccft. £mebium propotrionalc (nrcr ?ct frmium rquarrnm stbinter piimG ct fccunoum. (17 (18 1n(1'* 1“1*1|p|oj+| iJ |aMe| |»|e|f| l‘l Cfl^anccuooedma quarti menUrat (U<C0 fltljocljfc Wa» fEqoentta menKntnmbabena jcogutdo ptomprlo: babfatunrrpaituroetnenftrarto. CSitacuiufoincpiiireruallipiimustfrminosaPcfrctititifiwftcrriifeabfquaiifitrtnrbmeoiOff p2opoincnalitcrccnftttut‘’inr(T a‘tc.t5ic(i etiam tntcroctfintcruenircvnumpiopojtionalifmctJiS ^apio mi prrfcrtam ouarri gfj Utres minmioa fecunoQ pjopoinonc aa» bet argiimcntoj «:TiC6< hmaptimarccuriPipcrequam propottionalirotfgsobvtaaobetbapUTtbaoc.igitsrgaPltvta aoc.quareivtoaef.atftccrfrutttitpeni termini cfi git manifefhitn iamdTintcroctflnRmmirc mfipTopotrionalifcrmcPifi-Sinniinusicrgoper 'piceiimStcrtrtBnrfcmeticeagetUmctictnroeif cqualttcrfltcp lorceunoCl.cucccrgolmgbcrft'? pjouentantocfpcrfcprimam fefunoioaocng aPb^^rcaofvrbablt.quarccinrcrucnit^tpottlonalitcrmectHstcmtinisOTfiqSeratmoflranpfi. ^c:Supcrp3rticubrisintcmalIimcmu«numcras:iiC(5vnuonccpplurc9pjopo!tio= nalitcr intcrucniciit. «Ppiopiitio ftiperparticalafisait Puo equa p;uiDi non polTit. Scrpadum fuperpardCDiare. &efrrca minimiptopottiomeaa» b* dTibanc intclltafmuBi nollum ruperparticolarc intemallum in onaa ant plarw (nteMtallopl^ittt ^cstncP‘':tiooefcpf«abilccfrc.qucipoemon8rationcp:oinnjefubiuncta|^<rfaoleUqi)rt)lt. arSiraTinreruaUQCuperDarticularcquopoiuTOa-urincuocquBliambabiropiri^pactafl b«bc c7rScMS«numcri6fipoir.barfft.fit£pictirfoTtaaobtwbaoc.cap!opcrfotlquarnamb minimoscontinucBpojtietiakeinpiopottioncaap b-ct qz^aoeTtaaobcteaof «pk coui 0topTtionS^e 0as frt aao c. m osbfiHai rur fs3» feftfuperparttoilare.Srrcro quoniam ocffunt tres ilUa^abl^iniaminimi rSnSSf»X'l™»n^''nnoperp=.n,5,»arrtrnnt.om«,eep.lmi.Bt^o^^ narrtenlarifl ao f' Dcr oiffinitionS 5continet fet cius partem qucioco per mffinitient metitur f- £t fi^cTparsfltnuniCTUO cum ea etiam TJbbUofopbUwalcafevibcrivoluntnonBmpltuo effe cognofcanmrnccefTccft. ^^ tcmaHum non muUiplcj: binario muItipUccturtio quoo iTr ct fwc multiDljca rioncfift^mnlflpUiccft/napfupcrf^mflilarc. ^ Sciuteruallum nop multiplec neiyfuperparticuUrc. acwjplafnm InrcrualhmTaTqfl' quiocm dl non muUtpIer. C&tu<ttn»ll6nonmul^le.cab:ctouplamatac.lravieantp2opo:tioaaob?t>a9c.oico tnteriwUJ^8tnoBmttInf>le;:cfl«:necpfuperpartiailare.quom3nacp«mop(Ht':turiftulriplcr:erffoDfr Wtii p«fenM« abitir^anfimt multiplcj at pofltum cfln5 molnpE^ .£t fi fecunoo sixms a c fupcrpartlcuUre cfi fit aab byt bao ceririntcrualU Itoperpamcniarte mwtusojoj pwrtonaUe f^muaquoo j^p:eccc>mt£eft uupofiibfic. dt itica noramfi Intcrualifi no malnplcc coropofiram Intmiallum mtmmcaut multiplex aatfupcrparttnjUr* cfic. ct p*OpOtitDJtt« eSi ftcraaUum binario mulriplicctun atep lo quoo c? ca muWplicationc creabitur mulnplri: non iTr: ipfmn quot^ non erit multipl^. C^ddlconuofapjeceoentie. Sitcrgofupmotiapzopofirionlafisuralnduaioteruallfiac owi «e^cxcnpl^oncbabituoinia interualli ab«n5 iit acmtcruallummalrtplcr. oleo Itloem mter “tH!!"* ^5moitip^lUmfi abIntcrualifi multiplcj: tihefi acper bppotbcftm er Interuallo ab bln^o nraWpl^o forgat :ergo per pzlma ptefentia interaaUfi tcmnltiplexcnt. at pofitfi cH n3 maltolepergo 11 Interoallfi nonmultiplet ex aUculua intcruaJll binano mnlrtpllarloncnafcaf netpipfuffl qoo<p cx cuius mulripUcatione ottfiefimultipUrx criKqfi erat monfiratrafi. ca numero partiurupcrparticulariomtcrualliqucin vnum aoactctotumrcfhwut ct numero vno maiotcrqiiot intcrualli maiotco termini fimul: mmonbus limul acccpli0 refponoeant: coffnofamtur. bctrnampartifib.eaiglrorparsfecunofiruimemm c:eonninatruum totfib: etfito numerus vno coniinentbfertwmcfumptsettnfupcrparte8bfumpta6fecunofic.atquipofitfi«fieasrttunoSc continent bfompta fecnnofinumqt vno matotem c. St u^WnpUr2requiris oeclararionfTft aao bintcrualifi cftfefqualtcrfi: pz^tfionieni8totennmerftftimptt8,Ttfia80bntfefqmqumta:qaoni3mtnimtfcfqu|amntefunt b‘’£?r?a*aoK?fwfef^ muitmisquoni3 uct lo funt terminifefquiqutntc to a ^t> ndefquloctaua:quom3 minimircfquioaauefiint oet S. iocirco satautc licoritonomoooparttcularitcrfeOTWiiuerfaUtercont^laricuplsiovmuerfalircrverficrlt. C^ataquacun^pjopomonetmaioteotcrmmifecuoum numcruminoiumfumpri; cquifuntmmonbuefeainDumnumcmmmaiommfumptis. ^; I jSSTOU Islflm- I,| ;|4| tl^lSacE |<j7|,| .1 I»cill«ri:o I t|_^|_+ 7j S I pUtra- ['|'|'I'|*|jij +jfj « C!Srriraai>bque(un<|;p:opomortamaio:rmninu8:bPrromino:. &icoaruniptM&cun&fibniM merum mliio:c c«iuca efle bfumprte fccuncfi anumcematozcm.lflam per octaoam pzimi Britbme* ticca quoofirqzambzequumcnciquop fttctbma.BtTcroquoefiteiambzruntafutDpnffninpa b.etquoofztCTbinafnntbrunipnfecunoiza.isiturafumpnreatnP&bequaturbrmnpttsr^oum actinquibunibctaliteeacecd oetndflraciarptopontfi. '® CCum a!iquoim:rio:rc termini (>liquotitiitto«bU9cquifunKcavfiiuj5maiojuin at) vnum minonim rcpcntur,ppo:no:qiic7 numen nimotum colicao^umso miuotum collectorum numcnim. C8?ocfftfoiicrfapfcDfH«?.filijen-3pfrpzffep?f«nmfno?e9Collcfrif«iinbumniim<rfinialowm cquiruntmatOMbusrceuntQtiuniiTummfnozQeclIcrtld.Scb numcraemaiozficftmue tmiiinad ma<oz etnumcrue imtu'zu ritus rcrnifnus mtnoz. ergo eacemetlpzopezrtovninsmaiozfiaornizm mme:6:qucm;mcnmmo:i5 cclleftosaMtiiircrfimaiozirmcolIcCTOzfi. <t£fH*w5’bePficrltimccon tlnfioncofadlcofunrzvfiipfflmceerumpauloIot^rfozcn.^^erpzJmaCTimcerfiqacnonaeflzbabe* muofiiiuinciucmaiozapminozerefqunlecrellrttrinmaPPuoJOuomaiozeBequtfuntfrtbusmlnc» ribuo.£rfimaiozaC'niino:cfcfcuitmiuevt 4at>j:rre8matoJc8equifutitquattuozmfonbue^tfl ni3tozfcfquiquartusvesao4.:qu3ttuc):matc;e6fquifup^qutnq!minoztbue.Sirerom3iotferqu{ ocrauuo:ocro maiozcs eqot fuitt noue miuoz(bus.7 fra in eorimiltbue.?n mulrfpUcibus aQtn maior rupItiacftrtPuoruaovnHmtriiuematoz equus cflotTObuemitiozib^.St tripluszrnus maioz equus crttril'ue.Siqti3Oztip'uc:vnti0equu6enquarruoz.SlocmplU6:rnu8cquafa»otto. 5i»t“P«i'Par» nentibus rcroconfimiiccirrrr fi maiozeflrupcrbfrerrfusvr^aD j: tres maiozes equi funtquinc^mi no:ibuG.SifupcrfriquarfU8rt 7at' 4:quafniozmai02eocqtiifunffepf€minozibU8.Siwrofupcrj octuparncsrt 17 au y.ncucmmatozes equi riinrccfemtrept^minozibUB.eiboc pacto inrdiquis. Cfiprlnma autem batic infoperpaTticuUnbusvtenPlregolaelirimne.Sibuomatotea cquifunt tribus miijozibus quartuot at» ferTce ab houcIocto ao ouoocdnmos maioz ao vnum niinozJ ferqual ter cil.£r fures equifunraoquamiotzfcpao octo maiotao mtnoz^eftWquitcrtiv.Stquattuozttta» tozes equi funrquinqtminoiibuszauc octo ao nec^nnusmaiotvniueniinotCseftrerquiqaartus. Si vero orto maiozes noucmtnotibus funt equirmusmaiotil aovnnm mfitotfi cognofdFfeiquioftau% ^nmiiltiplicbus.ftvnusbuobuszouoabquamioiirresabfeTequiruntzmaiotmiRotieeR&atflua. Stvnuombus «uoabfcrzvnuBmaiotmtnoileefttrtplas.SivnnascquatmotsetPuoaooctozvnv maiozu ab rnum minct0fO quarzuplua.si vero vnueao octo buoabrebecunzmafozndoziscilocru plus.^nfupcrp.irMfrtbusdt tres maiozes equi funequinq; minctibus: aut ies ab bec&vnusmatot 3bniino:ccttr(:pcrbitcrtius.(?iquamic>:inafctc8reptem(notibU8:niaictininoti8(Craetrlqaart% Sivero iicuccqtn funt ab bcfcm tfcptcmjmatotmlnotts^raperocmparttes. £t in eompoAtisnit meris fimtle cfi-vt fi buo acqui funt quattuot biofmlbifimmsiqu^abmobom In buobue eoniQcrts fefqaciteTis.qiioma +ct rntiiobimibifiabibupIfiruntcffefquiquartfizirtircovnusaabvnDmbbu plus ef^arep rcrqatquartus.Si rres aequi funt ab quattiioz bet tertiSrniussvt in buobusfcfcttertii® quonlaquincp ttertia vnius cotiiicnt tres fmel buas tertias et vnam nonamipzoinbe vnus aeonn* uebitb vnum buas eius remae er vnam ciusnona. Si quattuor equifunt ab feitet quarti vmuszvi ii! buobue {uiicria fefquiquartis: quonia quarta vnius cotmcrztiqaatruo: bUnibifi^ bedtnani fcjrtam cj: quatfiozubco rnus nialoz cotinet minozSfentel eius bimibifi'*! ciusvnam bedmlfeptam» Si vero octo valent berc toctauamvnius vt in buobusfefquioctauiszqnontl bceem continent oao: et quartam:? octsiia vnius ab octonos vna cft fceagefimaqizarradbeo vnosmatotfi continet mlnozg fcmclldusquarta partctn/cr eius fctagcflm5q«arfam.£t fi copofitionls feries ptotenftoz euabtr: vt ftbuoacquifnntre):b'mcbtctai(vniHS etquarrcvt in conificttonemamrefqaalterotaiquoniafep cotincntbuosrcrtbimibiuvniiiseflquarraparsbuozfiietquartaparBvnloseftoctauapare buo? Jccircovnusa iripletcftbvnlusini^uperquartaatipoctauamparribconHnensJEtfttresequiront ab feptcni ? nona vnius quob in tribuo fefquitertiis coniunctie eocnlP.quonlJfcptemibis continent trcoz? trium icrttaparremzct nona vnius ao tree eft viceftmafcpttma. tbeo vnus maioifl continet mi notem bislciiisteriiamlct cius vicenmafeprimam^tfiquattuot equifontfeptS bobzami vniusboe cft tribusquartisetinfuperbccfmcfcjTCvniusmfin cotiifictioneRiumfefquiquarfo?: quonia fepri cotinent femclquattuoz ct tres quatcmar0 partesier tres quartevnlus ab quattuot funttrts bedme ferre:? bedniaferta vnius dlacquattueivna fctagcfimaquarfa.itdrcovnuamatot continet vnum minotefemeltet bobtante ?eres bedraaneetas at<p vnam feragcflmaquard-Si vero octo equi funt flO Tnoccim'tre8 octauas vnius rvnamfeeageflmaquartamzvt in tribus fcfquioctanispiunctiszqift vnbccim cotinent fcmcl octonanu ct octonanj tres partes:? tres octouevniua funt ab octo tVes fepa gcflmeqiiarte.?vnaferagefimaqiiartaaboctoeftvnaq'uing?tef!maboobeeimaJ)incfitvtvnu^a* iozum corirteat minozem rcmcl.'tres'dttsoctauae’ tresfffagcilmasquarfas etvnam quingcntmml ouobccima.fitbccabiecimnequovfuabarfiwJ^ronjvWinafinfequfttbuspatenttotmatilfelHoiq^ babeatur. Ct^ttplcj intmiallfi q: cuobuo ftapmie fuppartiaiUnbue:fcfquallcro atcp fcfqui^ «+Scsuplet mreruallum. *b'rfib fefqUalterJ) cfcquirertiDB. C:BmtabcUi«reattB:«aBi&efnfcfqi»lKf«b:b'»<rdftfquiteTrto8»Oc:Cico*9uplume(Tefllic. Caoniacnim afefAnaUer «ftab b«rgo per <>tttu3 buma puo aequifunt itnca b. et rurfue quia b Wquttcmna ao cttgimrpeT canocm trea bequt fantquanuot aetffee bponn runf ««obus ©Bot^toraequifuntaofluamioJ qoattuoinuniertwmmoifi«U«Mpeuplereftwj‘”“*"™"’''1.*^'?“l^*c cebeniimi» aMipkf ent ao vnfi c.qt5 erat pemolh-ancfi poteft tetiam ©emo^art.tfim «nl suo acaul funt quatroe: cimua a per octaufi pioloqutfi equna eflsuobua c. quop eni fora equa fBntetwi6qu(«poimloiafuntcmia.fltouotbupUfuntvntoa.igtturefvnu8aPuobuecequu8:tiu plrtenicvnuaqooofotffermonftranwim. CC.t ouplia micruano aK? fcfqualttro: tnptcynafqmr micruallum» 'jjj1 flcrrtplcptmeruallum. a1j, 1^j Bbsuplejcmfuallfi-bTcfefqualtcrB. frstntebcintaiwnfi:etaQuisfasbouple>;tbaBcfefqual^OT.slfoaelu9quoscqc|TipIicf^e quontSenlaiaasbsnplejcelbcrgoa per stfftBltjoneconfinrtbl0b.igitor aequatur suobuel^ctqi bfefqoalttr eft ascrigitarbeonrtneteet elue partemstmisi5.ergo^oftauBmbuiu8SU(^eqm fonttnbuacetSt»bequi erant vnl a-iglturttreecequicrun^ia.treeauttmtriplfffuntvniue. Igi tarper oodmionna atriplet eft mmetiquos rrat stmenurasom* CCuooupiioa iiitcrualla :quaojuplej: comunguni inicruaimm . 4j Jj,1 SesuplcxmtciitaHum a11^j flcgcmmarfisupleciiifusUu l«in)rSBobeqmfuntTnia.efrotfu8qibPup(«cftao«l^fur5uoccqui(^nfvmb.c funtTntbiquamJOfeequimmtsupbuab tr suo bpofltl funt equtvma.igiturcf qui^otcc^ - bunror vni 8.at quattuoj quasruplfi funt wiiaa. Igtrur per sceimam vnue aquaSfupler eft vniua c. ClCcrrninii rcpcrirctap que quotcungivolumueUiccatfuEpamcularcgaflisnarc. IgIfiascfefqualtef. IJBascfefquucriiua SatCTttfcSfit copofitua b-^oiUS enim accntincf cct rtua partem "8® atscfefqaaIter«ft.etqoonfabeonr<netcetelu8parfemrcrna:bmpemas<fcfquitmlu9.ct«iw repmu^lemtnoeasqql pttitifoPCitiartlcutareertrefuntairtgnatt.etbflclegescqutt)nmb« eStoperansftmneflentamSiSstfcfqoic^tauuactrefqumonussucortomiiouetrurgett^^ oetao^/aoitfca partem pofflDcnaias qu? tfefquiocranfi er fefquincnfi rue **^8"3UCT(^ct eosem quoeppacto mos© treaunoso qu attuot aur quotquotlibuent as eunsem terminum aiitgnapiB* —I—^1—II ' 'ttftaPeu*o^ferminfipeWCTW9l7ignarifeifmit^u6/fef4quartU8/fefquiq«ituet frfijoctauaa poeaturtnafquatnioi/quuHbctoetomfeinmcetpiooeniatcquiirdrcobabetmfe tcrti^quamm Quintict octaiiam.esiungo efuam tcrtlS compofttufnfu a.et Ibam quartSt compoutua ut Duuam oaintScowperuna fit c.fufl oenitp ocrau51ppofUua iit s.sice stmoftraiionerupcr^^^^^qoonla as c ccnfbfuUfuntabc&fcfquitCTtiua/refquiquarTua/fcfqmquintua/etfefqulocraauaiqueatmoBum pitfipa monftratfotmola. l >5 c:SiTcfqua!fCTOinfcraa!tofcrqmtcrtiumDonptumfucritmtcriialhim:cmq8rd(n' quiturrc/quioctauum. i *> T~irn 63cfefquaUer.£ cfcfctuiterrUxe. i^1»i! '1 ISaobfefquioctaima. CSitctcrniinu9flpqD<mBpjeccP<nti8fIi0narinnt»f<fquaItentbfrfi}otfrt*.fll>acref<iuaI«r* fubt>uco inrcruaUum mquiFtium bcrelicto intenullo 8at> b:quob &ico raefefqaioctaaS. qtii enini »cius quoo eO ccO fcfqualtcria fonel t^abet c et nae timiDiamparton. qparepv ocuoil^ioa mjc» acqmfunrmbu0c.'tquatiuo2afe):c*eroctoaabOuooedmc>cturfu8qmb!?utn8qucbe(lcfercp]t( tcrnuo eft.b igif l?abet in Tc ceiue rmfa partem.quo fit vt ceano£ocuui trea bequt fint Qttuo: c. et fcjc bocto cet noue bao ouooetim c. at oao aqutofequt erit ao euooeam ctgifooo ae^f fonj ao nouem b.per occimam igif apnnet b i cUia oaaujpaitem.dfa:p afeCquioctauiM a» bct^ofitb. is c^ntcrualli quarta touoocdmatTnam austertiarum rcftituHnt. qnatruot intcruallt quarretrotS cotnp!^ infuallS.t tuooedm eiQrt>{ intmtaUlboobecime: totfi iriDcnicoplcnMDTmiallu.quareqoattuozquanetouooccimouorednicaoumu2cqu3tor.oue igitur quarrerdtDuooecimie; tvna quarta tHbu80uoDeciml6fnnte4ue.quMenitotacqaar(mt;i coooimtoie.BtquaituozouooecimeiOuooedmouooecimarfiruntvnatcrtiatqrttvnatonuatcnla cm qiiioc ouooccim ille ouooecime cognofcunir efle ioem at<p eque, ontIgif tortua quarta qtribus ouooccimis equa eiTe moflrata efl ct vna ouooecima iUi aomnetattotiua tertiam partem reftttuunc. £qultur enivna tertia tvna ouooenmaquattooz ouooeami8.qfi erat Dem5ftranoam.£t^mia l?cc oemonflratiofcquctiegratia parrtcraKtcrfacfafttqioniatn moooplTmilimonllrarco2mminowni cumfcucp totius uitcrualU partemtjiinmc marozem partem cfFiccretUi aoiccta vna partium anficro qui cc ouctu oenoiationS vtriufcp parttum mafciturzoenolata-vt rotiue tertU pars mino: eft: ifctfa 4)cime maio::er fi ouo in tria ouicerta funt fcjcnccirco vna tertia attp vnafeytavnam reftitufit febam. SimiUtrr quo^ quarta tquinta partea funt ;ppme maioz an^ minoz :et IT oucie quicb in quattuo: furgunt vigiimiproinocvna quinta tvnavtcclimazvnl quartam rdhtu3t.ct t^>c pacto in fcqu2tibt»* vt Tubteeta mcnm-atfonnula. Certia cr fena vnam fecunoam dTiduot* iduintaciviceRms vnam quanam. Sc^ta cttriccfima vnamqmmam. Septima iqtiaoragclTmafecfioa vnam fenam. lOaauatquiiiquagelTmaretta vtiamTcptimam. IRonatfcptuagellmafcdioa vnamoctauam. 17 cE^uc p:opojnonco fcfquioctauc:mmo:c0 runtfcfquitcrno (nrcrualb* 11-111 Scouofcfquioctauieonificti. 1“1 b 1'i ^{SofefquitertiS intcruallo. ClCSrtnoa per feetam quarti 'aritbnteticee ouaafcfquioctauaa inter atezita vt afcfqutoctauv flt ao b/et bfefquioctauua ao c.ct flt aao ofcfquitertlunt imeroaUfi-oico aao cmlnua ei mtcrualiums 5-aao o.fllm enta fcfquiectau’ eft ao btergo ponauam l?u«us ono aeqoifunt ao nouem b.rco tq: beriamfcrqutoctaauacftaoc:perc3ocmoctobtiTifuntaiq}noucmc.£tcumtnn8bequuafttvniCT octaue ciuetergo nouem bequi fune occem ctoctauevniuac. Stqui nouem bmoftratt funt eqni dTe octo oiigitur oct’e aequifunt oceem ctoctaue vniu8.at occem toctauavmua ptin« octo fcmeli cius fluartam teiua vn5fe)tagcfimaqiiarti.crgo per ocamS buma vnua aptinet vnfi c/ vni eiua quarta» ?vnamfccagcflmaquartam:ctvnaquflrtatvnafecagefimaquartaperpzcceoitemmtnua fimtvna tcrriarcoplent enI quartatouooecima vna tertiam.ouo igif fefquiocianiminua funt vno fefqmfiw interuaUo.quooeratocmonfh-anoum. . is t£grea fcfquioctaui:ampliua funt fcfquitcmo:minu6aul fcfqualtcro intcruallo. 17i9 j5it 1'Sotreafefquioaauicdi&ett .1 !i"1 c 1 t. 1 ffSint abeo trea CMUfiettfefquioctauitvtaaobpzimuafit fefqaioctaaua/baocrctfioue/ctcaoo ^tiuazoice ptimo aao campUua efle fefquttcrtie^aift eni aao btbaocouo ffit fcfqmoctauuergo Dcrociauatnbuiueoctoaequifuntoeccmceoctataevniua.SeotCDmeitepaoopolituafit fefqut» oftauuaicrgoDcaoemoctoecquifuntnouemo/etnouecequioecemototiauevnivetoecemcequi TOioecim o?ouabua octauia.ct cS octo ocrauccptincanr nouem o:g per oecimsm buiua vna octaua cfontinet octauamo tdua ocrauam panemrboc eft vnam fcMgefirnaquartam. occem ergo cet vna «CTauflvnluacquarurvnoccimo/triboaoctauie/ctvniroagtflmcquanc.etpcrocamavivnoecim rea oaattc tvna fexagefirnaquarta ao o«o:ita aao o. feo vnoecira cotincnt oaonoa femel trea eo? [ilt i157 51l5tf iI!+< 1 1Y19i1V17 1b11kj ot” .tl5. »fit M«iufnt»cclm..toircUtlam Witc notum rit. ^BcmefitRrtnowTu j£0^<^iocta«aocdma p;opoitto:rcmitonio roino:c nirfue maio: cuaoft. effe mwditi tfCTa«:qua quie£at>6o tmmcro b/ lUtqs aggregatue c.tunc cat> b CScfQafnon«occtmaicflfcmitotuoniloJcmino:.^uofftTtfctmtoniu ntinusintcr m fcfqmo<taittDcdmam/ct fcfqmnonamDccimam conftitat coUocatum. —'- , I.et“ t< il.4-< I 15< St45_ b SSMOTJS»t»rcfiliiiioii«ttlmaj>i.otiioraniioniomino!tmlroi.Coitclnria*ov.otara» nuarta buiu* «wnw* P®^*^ CSonitonium maiusan cata cboioa conibtuac. I9a| cl I irSkabtcbf«nttoflamtno:l9interuaUum.Biuioocbfn «tepart» equas: qnlbns equam facio Srtmcb.i»w»bnounn?«wat.cTgobbtcbmttonu*.wnu>lglfOTatcnoi)bctcbfctniioni5 SSmafn abicb/relinqoef«b bbi a bwiU rdiqua pare tqoa ipcronltonlum mjnue r«Pf BffSirilmbbtabfeniitontummaitwcninbatatboiba (wg)poflraer8f)co(hturfi.etqu#ao. nioBam fmitenittm maUwcoUoalH ao pamtn grauioJem ita quo® ab parton acuminis collocare faciUmuincflb. CScmftonQ matoiteminimoc numeroo itpoirc. 104^ 11fP44 1 ic11c1 1t5tf 1!145 i ,a1 I b1 1118 11 1 1 1f11 esima bminimi nnmm^iioiwj minotfaetbedmaoctaua buioerepetti «Pocionariue.qm et ocrooilranoc bcameoctaue puioa foctonarius ptitmta cfl ao b.crgo fnon nomerat b.non babet l«t tBrboaaaampartem.bucoigtturftnbctaetcofurgantcorperfeprimamfecfioielcmenrosSritb. iBcticeo »ab cvt a ab b.cii igitur mfcr bcfemltonium mmue/femitonffqsminoaa InteruaUom.ct qt cb*»>«E«'«®ft*o»^»^®®®etbonnaerocEtem octauamduab;ctcoaccnutB«flaie.£5tiiictigiC tmuoeru» c/a parrm ciua oaauam.quare e c tonus.ttquia ton* ecfupcrat femitooiS minua bc tnrcruaHo e9:eryOCt>l!kintRfimrfmiroiii;ftnitods.Caut^{l9?rint minimi fad’eet&e«nma<ltHnu tcrtfl arirbmcticea cognoTcaa oifb‘a^^ 9ab etquo» rdiqau fatrit itn quotiea potca ab orft bocpaao ocincepe tvlocbiaa» vUirmim relictam vniratc eriti? moOo qut 3» Urua apparet ailtractio.quareper can9cm9ectmlquincam 0ct efune 39inutcem p:lmi. funt igirur pcTTiccfimam ciufccni iufiia ptopouione mintmtiquo» cfl p:opofitum. CScmitonij maio:ia b^bituoo :fcfquiqumraocdtnim fupcrai p!opo:tionc. 1187 iI1184 811048 1I3i$ S S1ciS 1b10 CSmrabmtiilmitmninibabiruomiafcmfroitamatotiaperpteccPltem repeTttn>ifol?ablra9iri€ flaobmalo:£etrefcfqulquita9ccima.fumocnlvtinp:«co^tibuaUmfepcfactficftperoeamai?uio qmnraoecimam partcnumcribctbccinucniruriccnrutrigintafcjretortoquintebenmeTntuaqAro «ouclocr0ooaebctfiatc:<ertfiefleaPbcircfefquiqu{tflmocrim3.otcmin(nnum^dl^a.cAcni anumeniebuum miua centli octuagint3reptc.c autem folum oafi milium centfi octuagiraquatruot ct ferefemie paulo ampUue.^Aigitur pcrvnocctma butue fcmlton^ matorta babtnjoo fcfquiqufta Pccima^JpomoncmaioteamtpfuperanatquooeAptopoflwnt. capotomcG mtcruaUum:minu0 cft fcrquiquartoocdmo mfcru9no.‘^tiDc manifc* flum cft rcmiioiig m«io:i0 pzopouioticiw inrcr fcrquiquintaoccimam 7fcrqmquarti occtmamreperm collocatam. Itr94 4. I1187 Ito 43 I14« 4( C14 ifl (bI0^4 I CSpotomeifcinifoitiiimaiueiiotmcireiamPijrim^SinfcrgoabmtminSmfcniitoni/maiotis vt ouo milia cciufi CTOctuagintafeptc aouo milia quaotagmtaocto. fumo per occima bums quarta cenmjpartem bfc3ouum niiliu ct quaptagintaocto qu3 tnucnio enecciitO quaouigmtafej: ct quat* ruo: quartcoccuiiequc fu o.aooo eam quariaoecimani aob: et fiat numerus aggregatus c.miic nu* mencao bfcfquiquartaoeclma gipoitio efl. at cmato: numerus cft aTiquio^ ouo miUa quamagita oefocteenta quacjagirttafcyctquattuoiquarteoccimefuminaftmulattollantciouum miUuceiuu nonngmtaquattuotietfcrcquarrafnius.etafolumfuinmafitouum miUaccntumoctuagintafcptc. conftatcrgofefqulquarcaoccimamp?opouionemaio:emeirefenMtomomato«;at<pp:opof«um. £o:reUnumvt pteceoentium notum cA. CScmitonmm mmuo atip Terti itomum mams fn lupcrpaitj'aihn'pjdpqjtiootfnon caountifco ca in fupcirarncntc ranonc confiftcrc ncccflecft. ClBam fcmitcniC mutus per co:rclanii vuelTnuptimcbufus caoif infer fefqoiocrauJoedmam et fefqmnonaoecimam.arqut inter fcfqiuocfauSoecimam cr rcfquinonaocctmi nulU caoere valet inter cepralmcoiaqjfupcrparticularis babimoo. funt eniillefuBparikularespwFimcmalotattpininoi tffirur fcmuoiiiii minus in ftiperpaniculart ratione non conilfHt. ne«5 per toem femironium maius. THam percotrclariQ ptecco^iscaoit in aliqaa j>po:iionemeoia inter rdqutqatnrioerinu ct fc qu{ quartaoccimamouaspjoFimasrupcrpirticnlarcsiquefupcrparticuUrc meolam nuUlaomirrunt nd igitur fcmitonili maius caoit tn fupcrparticuUrt ratione, feo et com rationes feniitonu minotis atcpmaiojismuUisfupcrpartiailaribusTtiamvirumeftrmt minoicsieriitctiamminojceratiQiie ouplariqucmultfpIiciumminimaeA:vtquca'mmimonumcrooenommcnir.nuUu6enimiiumcruB binario mino:.Ro efl ergo feniitoni|mino;i8b9bttiioomultiplet;!riritcrneqsfemironii maio:(s.rdin quiturigiturufufficientiofuifionccum illcfint inter mafo:eterminum|7mino:e: easdem iuppar tientlgencre.qiieo eA;p:opofit5.CE?oramcp:oprcrfopbiAa9aPuerrereUfctq>fotiu8tfuepaitcs cofonantie/i confonartariim partes infenfe; femper in fucpartieulari; fuperparticti aut muUiplui babituPtnecaountiremifrcveroinfubruperparticulariirubruperpartfcriautfutmuitfpiici-Stecro et fi ita fiurolemus tamen eae omnes PicereefTc inrupcrpardculan babimoine faperparnenti 1aut multiplici iPemruBparticularectrubruperparCieiilarcrcputStcsTpio vno computantcs.ftmiLitcr riiperparrlensetriibruperparriens/tmultipleectrubmultiplcr: C.afeuncutn comma mcboioarcpenrc. /af16It b 1I5tf t45 1 1t> e( CSltabcbomafuptaquam fit ptopofitfi reperiremuficficommaiin qua abercb fit fcrquiocraua ptopottio atep tonus.fint pzeterea pcminimi termini femitonii mtnotts. Piuioorpacili abin parica equalcs fecuitpfi oiej: quarum numero ab Ipfo bverfus at capiofeconofi numerfi eit in tcrmmocaru pono notam f.tuncqueptopojtiopao c:ea crttabaofb.qu3reabapfbfemitoniuiiiinus.£lurfu..t cbfpaaumfecoinequasparresfecunPiinumerome tearumpattiunitbipfobrerfusa merienoo; fumo fccunofi numerum p:« in carum termino porto notamg. mamfcftiicnam cAg bcr cbelTef&ni cCommotinmWmienummecenlbtuere. 155l44t ' 1b1 1<14188 I1 e11 4J>7««4 1 'b 4.7158» [t t i 18555« I>tftlo 8I59049 iiII 0'' f15« 11»45 11 ifi i)b1 ]118 1 11M^ ! 1 1 1 a C£i<irabn‘Ui<in{num«rifrmiix>nflmln<‘rie.&ocoainfcTlnb;etb{nfttcyurgantc©ft^tcpc» ^5 I441 t©ecuofeintt«iiatnIno:ac3iuncraJltpt<tfreafocronflrtB«<juict®«}m3Dctau8puiuecI(p:imn8 5141*8 aeb.quarepCTvnocdmitf^aritbmetKfsetprttimaaoc.nenljabctfgiturcocrMam.t^coergof 7*55 octcnart6mcce:TpJODfnlant0l7*^*ft3onarure):fin<4€mfieft<ti8mtntcr0!?«tpt€eccuofc5 tII? iBiromamino:a-etq:edlt>rt8aywr8{acw«<inmuljfitqjeomm8gCTCMtneli.wbifiinillomfft 79 sUao(cflefcf<\Biocraiiat«Ugdreptopo:t(onon(ioaf<f<luiocrBuam 8fo2dfgpetptPuobBerrtnis 517 t«iUemin<«tbt».funti^tuTU0tiainff{c5m8na.fco^antmintmi«o«imaqu^tatm(farUbnic» 9ttcc8nonimcft.'^mfubtlracfovt«8fpf8PomgabU|«eoqnotreUctfieniubirraftoqDeappotcfl 858 abg.«nK»«iieq)«wii»3waPtfltuaa»<«rtapparmrdinquirurwiita«.£tqaeaP|iieeumcomma tI«oabuaWeUbaacftpicpontumtitaquocp^aaUtm&cflcoinmaouabqsPiertbuaaibiangcrelaiu 5puaboatnrcifcrcrr. 'cScfiiiiifcptuagclIraaquarta; ccmmatopzopomoncmstircoroinir. /^5r44T )55157 ». 7» f5i4»88 (70* 47» _l Ig^ 974 Ibic7-4. I ' ^SintentabRiuilmi numen coniariepcrpcepcnre reperti, elicto per oeciniSIpuiuarcptuagdTtnl qutrtib queiit c.aoiiclo itaip bet crtmu{;-i (oalrfcat aggrcgecur<9 numerue r>. qui fi 30 bconferat copcntur^efqaifepruagffirnufquamie-atqul it>c Dtmno: aefTe otpteJjitiiwr.fHpcrarcrgopcr vntx cimam buiu«ratio comatiprcptuagcAmiquartam pwpomonem CComane cwtino: cft.vnoc fit vt comario ^ ratio inter fcptuagcfimaquatta ct rcpmagcfimatcrfla conftiiuta repenaf babituDine 1551470 t155 »4 4ii5a4»8S I718» »I I075 )SIbIc75 i CSim8bwpiiu8miniminHmfric5in9rt9.capfofcptnag<ll!n5wtiampart? bquefitc.iffgobtc nmal«aggregm>rb.mr(poa&bre(quir(pmagcrunufrmtue.ec«maiouerreconrptcitur.ergocd matierattciftfquircpttugeninatmlaptopottiooeminotefl:quooeratnjonnranbom. iCctr^ruimvt atianonmeft. CCotnm^ratio: in fuperpartiente ratione confiiht. 5, CnonnliofuCpanicuUrKSAftetqui&oquioetiaerucparticuUreeptojrimererquireptoagcrtma qttana<tfefqaireptaag<nmat<rtia;omn{p2otrnBmcol3m;dubantfuppartutil8rc.itantorninQsin giKmnt^lkic3ftnere»8lcbit.wquereptUBgiii» 9na8 fupaT)arttcaUr<«l?abcatfcraaioz4:8.rcliii» quitur ergo Bt ea intuperpartiete genere confifter«po<Tit.cnen{eamaiozi6ctnUad2i8l;abUui>o. ClRanonee fcbifmatie attg wafcbifmati funt ignotc/atep irrcnalcs.quo fit vt qua? « mimi nfien tcmigonicfi latue no babcar.mcotciaiii ronco ignotcrrronalcfiip fint oco ^ 1158 1145 I1'i 1 1fi (b1C1011«1 1551441 15»4»88 1)1J CfiMrationeB ignotae 9t<p irrarionalrebtcimueqDenHllocertoicofHntoepnumcro cefignari vr lcntta^:vtncqiingeomctnC)Dia(netnctcoflequa9i8(i^pot(io:quale8rc^irniat6ioiarc^irinat&^ II ;vpo:«onc0 c<rc©i£{m*.Sfntcr00pm0«bm{iiiminumfnfemifen£>mint.d8:c5crtm5<x(at:abui^ mu^tubenfvtiUicquotbnjoftrarfi fftnafoffrnoucnanoinrcTttrnariotnpjcrqctlJncutnartlinfc atnucru6c{ctouctuiiouenaruinffctargcbat;fratquat>jat" «Ifrnanu6n6cnquapjai*:frsopo’ occim.inuartarctrfartfl>menceobmin"fcmitonunitoJio crtrcmfictcuctm^peno qojaJim 3paru piouem^a: no cft nunifrue qoaNams .(i pofTiblk cft ergo vc oiafcbifmatos ^>po:uo in numeris fic notaifcntcrgopcrfwSquarnaritbmcnceeeuootafddfmata lnmininnBct>c<6mntta.man>frf1G ciil cnrumotatcbifntaritranttcnumkiiscimtctGcoctDerimulcflercitutoniGminue.tcase^e fcniironi(min«ia tntmiallfi.fcoct cumppcttioncsceiDefintcotinuateiiiminimts ergo per quis wmquartiaritbmcriccsccfuntinfus ^'Pcrtioncminiitii.runfcrgominimitn^rpoitioncfcmijoBU minoiis.fcccttalesponnfunta-rbngiturcctc itoem cruntnDmCTlcfiatbfc5ci&finacteio<mb iptjettTfa qz que preponio cab bea efl oau c:ergo per pdmS feni antbmertees enumeres cft quA-. fjatus etcnomcrusqiiab:aruB:quarcTbi»tmnDmcroeeriam quaoiarus.atbbemolirafnscftno quabzatns^itiratptcemnumcnisquaoiatueetnSqnabzatus: quoueflimpoiribile.neitgirurbus fcbifma note jjpottionia habebitur, ct iocin ue plopottioncfcbirmaies monflrabirar. flnicnitn ab fubterbdignanmminiinumertcomatisiqtperviceflmloctauambuiasbmlnojpjepojtioniacofn* tnatio terminus (it ct ouctu octonarfl (n quaozarG (areris «ucftoium qofquagmrafcr.et octonarius n6cflquabta tus: ergo per tcdmaquartam fettt ar(tl?incti<c8bmino: rerminus comatis no qua Otatus.n6igiturftmiUvrp:tod0 partis ucnioflraMonc ab certis oc(Ignattf<® numeris potrfl: equis ptopottionibus oibuci.cft igitur fc!?ifniaros cUis rc3mebictatts ratio ignotaiat»? irratknaJis. Coi rctarium c% moto cctnoiirationis notum crt. Sonue :cuobiis fcmitonilsmmotibus/crcommate coiiftat. Clflnm ratio fcfquioctauabuobusremitomisminotibusatcprno comate fonllat.rupcratcnttuaa ticfcs tuotBrcniitoniaminoiavno commateiac totms in catem ratione conriiltt.col>4t igirur tonus ejttuobus fcmitoniismtnoiibus et commate. C^omis aDuobus fcmitomis miuoubus: vno commatemftat. ClHamfnbfh-aeto afcfqutoctaua^tpottiotic vno comate rclmqufitur tuo femironf amfnotadgimr ct cotem comate atono tttucro tueticfcs ttuo femitonie tninojarelinqufimr.ttflat ergo tonus a tuobusfemitoniis minotibusvnoicDmmare. CBcmitomum mmu0tribu0commanbusmawsdhmmif0vero(j»armo?.vfloe mamfcftum cfl: apotomen plura quattuo: ct paucioia qumt? continerecomata. rX77^)t99>t I5t58 j0ltfii 5?o 74.70 <» 4.t il q5i8it 8589 ++ 5SP 55 iii 57iSf 77tfi^iTrft Pi(C 3<Jooo<i 95 t4ot 8oi 5tfojii5g9ff97^,rf 0tl 78JP1S I79tf5l0fotf ltf7 4(f5ffl nil55tf9j965075?S77<S#5 to747 mto<f 78 95 45 o^'8 79 <J 48 ^l 75 l904. 1«70 98 tfo 4.85 ?I 7tJfi It 6I54 li 771 8 f11 9 I40I£3 e11740 19S4 t> 51418 8 c551441 b al 5 4£1Ronegrecacuriorita8caIcuUIaboteteterritaquo minus quot comata in tiefi quot in apotemc quottenit^ in tono finti perfielHgaret .quoo niti ^lioiibustetatum cognomnem cd it qnocp plus labotis^Cvcml^ivirumcfDinmufidsmoBulatlonibi^vtusfvnlitatircpafeTatmutrDjfectltcm.qui tamc It cognorceocrtteranerinthoc paeto oepteOraent.Smtabmimminomerlfemttonii muioiis et ct minimi comatisiper teejmSfeptima toedmaoctauatn buuis rcpertutuco bin cet 1 1 ^uciitac efet am.c tveniar g: per feptima ferenti arithmetices fat eeOeSmatis bsbitutolct per octauam eiuftemgatchabiruto ttefeosiremitonuqjminoiis.^ntetucoelncrftnfetcmgtnarcanmr hUliperfe^tam quarti arithmetices per?fsdle cognofd poteflli hf^ntiiicrc tuo commata.ctper S2’ ftptimjmfecuncioutbtUeljeflreffwitwUammcnus.tdnPcsuio&mb ftflnlmoinl.tcoM Mnf tojunt mn0:per cauptm fertam tjoatTt cogmra faciUimn rft nmcontinere im cemtnafa. « pcrlcpftniamfccuiiwteincotmeTcfcmiiciiiummmua.fltnni-mcrueccfincfiitiircffciiiiMJo.trtO 0a? tn ftmitoHtBmminuetriavtfKiiletfuperattptonimafa. PeinMfUfobm bctUtuuetbiBU etruoo:9meeTfBrfi5i*oaBn«TC>pqr.mantft(Wtftwnt«lD p:tui»q pcotircrcquattucuomata etrpc6iinerefeniitOBifimmu».ainumtrn»rmmoJmnrtnfroq.ig<fur quaiiuc: comata aitipUu» fut» femitoaio «mi«e.£o.TcUnfi flutnn b«nf »«6 ftmitoiiifi maiu9 folo cemate fuperat rettU toninm miiU9>stq<ii femitomfi mtnns plura tribu* xpauciota quairuo: vt moeo vifum elt coniinet coiiwta.lgitur vnico fupcrawectc comatefcimtotufi main* qitoOTOcaiic apoccnic;p{ora quattuot t paociMa quimp cofinnercrft neteffe. Cffcmimiplura reptem commere commata nccdfctft. rWamrcnucerfenutonioniinoteTapcremeccalefritatqftSlhmitor.aeffmironifimtnueperpe roinmimafcntinetcoTutaTampliosiftperptccttenr^apofcmeqaariucjetampliuetnaaufcm «qu8tfncirampliB«:fcpi^ftjntci3plioe. igiionona* plura 5>ffpt(mc5tinct commata. CSeainm^lemmtottHn a^nRcaUum Rnia. Jf Crqmtonuc: cf^qui tonum ac rcmitomum mmusconnnct. quem ct rribcmitoniu mfcmuoniufp: infmus^iccmuo. &itonu«cft quiDuoecoplccnrurronoe. Srnronus Tcro quitrce. Confononne fimpUcco funt twatcnarontWapcntc tDtapafen» Cempofite vcronMapafcn orapcntc/bis oupafon. riarclT3roncftcofonannj:quccr»ucru3llircfquuatiqrarioncn3fatur. Uiapcntc :qucnafcnurctfcfqualterj. rlapafcn rcrciquc cr ruplj. Tiaparon ojjrcrfcrcftqu.iarinuifciuncfffonnifmTrPMpfonacPiapcfcofcrflnc 33io cupjfon :cfl quam coniuguni t>uc onparcn ccnfcnjntic. Cftw altiuG afccpuni t^rl^5age;<d qalriii* afcmt-ctibua rcccs quo4 pacto dlt* fh-ibufe vife ITnt iq-vmcuHpfermefuevcciamobUilimitefcBaDconfonifiambioaiapafonnaiurafcccrit q-q; babira ronrcpUiione muflea abuJcb cofonannam bie cuparrnirtliquSnbabraturq farfll:m.f puraiicrunt vrquiaofcr artyquarerMapafon mnflcosmcPulceaptarereluerint.etbscqfcc? tccaufantnfnt fcrmcoeambefinicbaiOctcrmmabac^aicptraoecatnfcipIinamufuaKmiffcipTtbagoicnorranr» ccnoliKpuratee due limitibus cotcntitct pjtfcamiwcreqj aatotitati fecu ti fuftttfter bererniifiafle, quo» tno* tn b« Utabimu*imiran* CScfquitonu0 mta rcfqtfiqumtam atepfcfquifcrram coHocafuc cfl.^vnoc fit ncriT fcfquitonusicoinoc/fuauitcrtp auoitu fcnat:nonDu tamen eonfonana poncous fit. ‘ Il9i 31tSS 11S331}a+J 148 3f40 s It> 51a1e6J_b _/_c 1fS10n1 CSlt(t^A^oc^f*<^magtnrioao:bTerooticmraquiquagtnrafer;etc»nccnraqua»}ag{ratru. per »edm3octauam fecfibt buine manifcRQ e(l aaobelTe tong; tbao crfTr frmitonifi minua.qusre **»c per »iffminoneentfefquironue.quc»icoln.ppe>:none mincte confiUerc^fitfefqmquuiulet matowfefquifetra.Wam capio per becntiam feomot butus qumi3 partem cetreniet numerus 4.8 et tres quinte qutfitf.a»»otgiturfaocei aggregatus fiat 9:qaimaiotinuenttQr$a.{gjtur per mre> cimam rccflMbBia* » a» ca>pottto maiot cft ^aa» c.2tvero »soc fefquiquinta eft.rh ttaes fefquia ronu*mjH*otnoiienttnotec^rurus<l'fitrefquiqoitap;opomo. £apiooent<9pcrcan9j»ccimam fecuno! fertS partem cetvenier micbinumerus ct femis qut fu g.aggrcgatuo igitur gao numes crca(fuatRumerue.ccrtumeftnumfrflcmmo:eefre5’3-quarevtpnnsa30cfcfquifonuBmaiotcft q-ejoc-quiinferquifcria a’P«Uonecofhroirur.qui)9ellp:£>p<>fiiu.<:ojrc!ortatiiimcognorcirur.? ptimuq-feequifcnusfuauuerfenaraocirumtcuiuflibcfmoficisinoouiaiionibuGmteti fipem facit fenfus.q’ vcronobuin cofonantiafintccirco euenit qfcfquuonus in fupcrparticuUrl rattene no con fiftitiqujooquiwrm inter fcfquiqutntJTfcfquifcitSptonmasfupparticuLjrcsiiulIacauitlnrcrUe*' mreiatpfiiperparriciJiario b-tbitubo-nec? quiofcU in muinpHd gcnerctquoniS per vnoeciinJ ptiini butus oapla .ppojrioquemulttplutftmmtmacftieirfefquaitera ctfefqaitcrtiajjponioncmarimio quib^ruperpamculanbuoicrurijit atep nafcitur. coTonantia autem omnis per bimnuione iii fupcr particuUnaucroulnpliciraiioncfonfiftit.dlergototumpaspofinjinnoium. {II ^C^niocmoitonusmtcrfdfliuitmiamatcpfcfquiqiiammmcDmsrmmimimuficam complet arepperfiat fwimomam. '85 i/ lft{ 'jo; ,71' 'tf4‘ ^y> \'B\ i£ I".gi ic iif ?i .6i ''• lEbamonii rr (ORfonartam iocm dutmas rt buluavt p»cet>^e p:oct9tt ocm«nArtno.&nt ergo »bc©U{>romwmiiumiac«imtuti.vt8i|7i (54.&1CO plfoiiBm«a©cf«mftflere»ipjopomftnemu)jc fcrqufterucet maioie refqatquartat niufitam c^ronannam l^ano qua$ ptr<iCTrt.£apto ent tmtMi partemeper Dcami fccunot butus vt frpe factum en. et venit vnum xvtemtf cum triente vnus8:qat numema fit f.abbo iraqj fn«ment? a» numerfi ctf4.cr aggregan» eft cnm terti» parte vnlues quiiocmUtD.mantfefiiicnuaPcelTererqDlremii.ttbmaidirnatcBnruittentaMimraratvRumet cnuagtnraro vero ocmaginta quinqj xampUod.cfiergo f^uiterctaptopottia ntono ntalet. t:c mr fum capio quartam partem cque fit g: quam aooo ad cet furgat r. qui erit a& cfcrquiquami* .at a maiot cft c.tgifur per moeamafccunoi bmuettufonee fcfqoiqoaitfi tranfcenwr.et cnm inter fefqui ternum xfefqutquanfi nuUua caoat ruperparticulana meoiua neq; multiplept ent ergo nronuu in ptopotcionefupcrpartifrtcoltocatue.qQaremuAcamconionleumrctAia mafiaamoduUcionttmg fit cupl?onuB fuauitcrqiaudimm fcriene) nodum tamen perficit. jGDiiont mtcmoUum :fola rcfquiionum fuperat apotome . CTHamfefqoitonusvnum tonum cotuicf integrum et fecundi wnicortnetfemitenifiintnoa.rco ei tonueCTfcmironio minoze tapoteme confiet euaoatt? co8Utu8.crgo fcfqoitono ao fecudi tom tom* pletionC foladceft apotome.at ditonus folo» duoa incolumee intcgrofcppofTidet lonoo. ergo mteni interuallum folaapctomc; foloqj femironio maiotcifcfquiteni fuperat mfernallii.qiJ eft pwpof^ 4CDiatcnaron confonannam moata cbotoa collocaro. 45l f1 ) ac| 0l ej «»| GCum enim epitritflifefqattrrtlaqi ptoponlo: cgfofMntlamdlateflarofl crecn iedrc»»«tqa«cfi<9 cbctda vrab eam in quarraot equas pomones dtuioe.vtac:comc:Teb.etdteoaba»ebe0aiboar« diafcffaron.lRam ab continet eb et infuper aequetertie parncbequamr.efiergo wtenuUum ab fefquitrrnfiadcb.ergoabadcbperdiffininon^cc^onatduteflaroBictcdfoiuntudiatcflaronut cbotdaa bd3t3:coUoc3ta.quod efi ptopofltnm. ^CSrntonuo:conronannamDiatcfr3ron tranfccnoif. CWampcr decimSoctaiiamptimitTcsfefquiocianepiopoittones ampUaafuntfefqoitcnioinTfr* uaUo.fltqui in tribus fefquioctautetpcr dtffmmonftrcs confillant roougitor mtrbuafefdocraols confifiir rrtronus.e in rpimro/rerqutternoi]< interuallo confifiit c^fotunna dlareiraroo. igtttir ctfer nantiam diarenarontranfcenditvtptoponebaturicnionus. 5<rCConfonantil oiatdfarontouobuo lomo arc^ remuomo minoro ccmftarcncfcfrc clI ^nocfacile compnrarum cftfcfqmtonum tono/ ct Mtonum femiromo mmorc citra oiatcfraronconccmm ocficcrc.flCompcrtum itcm cft confonlnam ouicffiron quincp Mcfcoci DUO commata continere. tMarenanim confonantia la ei 11b|_ C^itabet cb confonantia diatelfaron-dico eam duobus tonis xfemftoniommoieconfiare.lRam cum ab t cb fit diarciTaron^ht 3bet c bper eSuerfionem diffinitionis fefqmtertius.Bf cum (eimto* ntummmusper diffinitioni fit id quo f^quitertiabaobus tonis maio;efi.conRnet ergosbf cbfe^ mitontum minus etduostonos-igitarconfonantiadiatenaron duobus tonis/fenttonloqtmiRoie confiat.fft piimii cotreUriii bincfinlc cognofcimr.Cum enim fefqnitonus folS tonum xfemitontfi minus (ontincatmcefilgttur ipfi adcofonantiam dtatefiaron compledam per pzefenrimus iotiuc • et cum dironos folum duoscoplecrarur tonos: deerit ipfi eandem complidam femitoniB minus. Sccupumvcro bmc Ijaud difficile pcrTpicitur cognttfi.Vlamcum tonus pCT tricefimatertiani feciici duasdiefcstvnumcommacSrincarduoioiil quamiotdiefesTduocStincbontcSmata.atpcrpte» fenrem diarefiaroR confonantia duobus tonis vnam diefim fiipcraddiMcntinet iguucconfonauiia diatefiaron quioi^ diefes art? duo commata.quoo efi torum ptopoftrum. 85 tom:Duaa watdTaron confonantiaB wocommafcvinaini'cuaDunU9 m3io:f0. Cl^‘uraDtiBrtftroxcnu« munm«Mdteffaron cdTonantiam Duobus tonis rt intogrofcmltonto cen ftsrc.t pwtnoe dumoUtc^aroa cdronsnriast^atniv tonos Ipim.cnius rrro: rt rrrrto mulkrs etoi 0<atTtai bo<t4 et rxb«<r tpucrv^ facile caumdmr.Dsm prr psrcev^tem cofonanna rtareflaron ni c[ ouobos tonis et frmlroitio integro conlUt: wrfi er ottobus fotus tfemi tonio miuoje. qaoo es viceflmapttma (MoimtcrrefqaioctaujMctmamctrefqiunonlocamlpwponipnfcoUocatur.mte grem aati femitoniS per oraindquanam rturMmqBcer rcTqturtsta occima 7 fefquircpntna Decimi coUocaretar.Dtmtnunuseflergoremuom&m>nBS iittegrofemttonio.fiutarrgoTtpcrpteceDetem Mofi dhcifMunru DiarriTtron duos tonos et ronitonianunus corinettoiie tgirur D<3trtraronc0ro< Baatiecfinnebunt tonos quanuoret duo remitonu tnlnota. et qt prr triccftmStmiani frcfioi toiws cotinet DUO famtonla mmota et vnum coma^go oue DUtefTarofl cofonamic qumq$ tonosvno coir< mateminurcinnent. qumqt igitur toni: Duss DUtrlTaronc^fonaRtias mocSmate vincit atib euat Dniu maioics.qao9 uit^Dcbatur. CConfonannS oiapentean alTisnato nemo ccnilimerc. (5 :i (Iii^fonitia Diapite » Jfi IC [0 ib dSttaflignatus numeras abrupta quiiciTamlltcdronantta DupentecoUocaritDlnito ab intres SDittout equas partes prr notas aco b:ita vt abtixs cSttneat et c bcarum contineat ouas.mt prr DifistnoneaWocbb^oUfit rrfqualtCT&cp intcnialli-rro cum eorenitia Diapctcqtrr Diffinitioni csratnternalU ratione naftai-r^o abaoeb ciConabit Diapete. ent^abaoebinDataci^toaalTi^ ^etoucneruoeffonsnna otapete collocata. CCreo loniiconfonanna Diapente trmue funt.7 quattuo: «nocm confonantiam tranfccnounc. C^tfierquinta bumsfarisfcgnofcafrrifenl; no pcircfff:cereDjapctcf5fonantia:bccciiaonrDtt mronum Dupcrc confcnatia rITr mtnotcm.lHam per cccim3oct.iuafn ptimi bmus tres fcfquiccfant minuofont frfqualirro intrruallo .ct per Dccimiiionam ciijfDc quaimo: fcfquioctaul fcfqualterum ruperit infcnjaUii.£onronjria ait Diapetc mfcrqualtero Hta rft.crgo tres toni in mbusrcTiloctauis conf^titminus funi coronantia oiapente. tquaituo: toni mquattuoi priftcntcs rcTquioctauls Dem conionatiam magmruDinetranrcfountuiuoo cfl totum ptoporuum. CConfonantia oiapcntcimbue toni© Tcmitomocp minotc conftat. fit vt aDia pente fubDuao ronorDiatdfaron confononua rcUnquaiur.fubouctaaure oiatclTaron confonantta :relinquatur et tonu©. CfUmper Decimiquintam piiml (t areTqua' tero internatio ferquitettifi ocmptfi foerit :relinquef rrfquioctanum.reD vt in Demoftrattonefrste butas viTum eft: Icrquitertia continet duos tonos cum (emironio minotexrgo coronantia oiapentererqutoctaufi tonum vlrra duos tonos cum Temi^ tonio mtnoie continens :tribus tonis ct rem itonio mmoteeonflabu quemaDmooum ptopoffrum rh. Coneiariu cognorcitur.^ftam Diapente per picTentem continet tres tonos cum remitomo minotc. at robUracpo tono rrllDui Tunt Duo tottiiartp remitoniiffltRus. et per rescam buius duo toni c6 remitoc olo tninoteconihtuutu coronannam DiatriTaroiurubltracto t^mr tono acoronantia Dtapenreireitn: qoimr Diateffaron.rcD rt cum Diapfte confteres tribus tonis cumremitcnio mino:c:ruDftracra ergo Dtatrfbron coronantia que Duobus ronis et reniitonio minoie completur :rdmqnef qucaDmooum recttiDa parscotrdarff puponit:tonus, quoo efl totum con^anum. C10iapentcconronantt3:inmu0octorcmironuemmo2ibu0 connnet. CiUmcum tonus vnos Duo remitonia minou et vnum coma corineantres toni tvnum rcmitonlfi mmasreptem remitonia minota tma comma»cotinebunr.ac tria comma»per tricefimiquintam recunDibaiusremltoniommo«rantcontractio».ergoDiapentequcpcrpteceDftcmmbu8toniBet (eatitoniojmeu conftatnn tnus octo remirantls mlnoiibus cottnrbtt. feo qu&Dmooumfacile mon llratuin dt Dlap«mtec6fonantiaHi nSDum oaauura attingere fcmitonifi mmus/octauam Dicfim*ita quoip facile monftram cnetieanDem conroHittam noDum reptimam attingere apotome CPijpenrceonrofiancaioiwno/rcrquftonotpconiunffitur. CBsmDiapente per pennltlma tribus toms et femitonio minotcconllat, ct oironos ifeTquIfonus flmsl tres tonos et fcmttoniO minus effiannt.igitur Ditonos attp fefquitonus pariter eopuUrteofo nasnaiti Diapente cungunt:quoo intenDitur. CConfotiatmaru oiapcntcT niatcflaronitonueoiffcrcnria cft. quo Rt vt DiatdTaron cotifonatiiuaDiunao tono: confonantiam Diapcnrcrcftituat. HI (tC'ifferctta bicvoctf ea fpMio <m» mate: rapfrttminorf.lUmcc«rd«rti *ecimcb«lw fnb Oncto tono acofonantiaoiepiente reltnQDif (onfonitu outriTaronJ^ i^umlbMio coirfotuntu DU* pcntecofonantiaotatdTaron ed matoi^ tSitor prr oilFumtotUit^anim cSfcoantiart tetna ti4a, i co:reUri5 (Uhmc?;}>wpofltiono notum cH. ,.«rCtoDiatcfTarwi: fdqmrono confotianriam wapente tranfccnott. C^arcflaroncnicrmjl^tueouca tonosarqjfamroflifiinmoo cdnBrt^gobiaourrfTcrSi^at taot ronos er otfo fanitonu minora 9rtnrbir.aTqui aqoattnoi timta tMtobaafnnuonua tnuoub* octnpto refciattofliKrdinqufirur freo roni et frmitonifl Riums.Sn}ero^ n«tnijbulaacoTotuiuu Diapfflte tottoj tonoa c6 fcmitonto minote copUaitur^go blsPUtdlaroatfcfqmtono confosanttl oUpeBterraitfgTCPUur rranfccPiiqjqufapmooum ptopomtor. »5 CCofonanncoiaidTaron ac Diapente «n maomwftiKaartoiIartbiw ftitit cottocate CVametPi<Fininonec5ronanttaotarefrarontnep(mufer<^iitmai0a>po2ttottecoUocsmr«oia pente mt^tolia atip refquitertia. atHiuUefupparricuIaresreTqiniltera YfefttBttmtafcwt tnalote* ^ama^ecfio8t^e^tUpa^re^^mattmefunrfe^ecofeqo^fc8patTea;pc^omtn5nItJs^ntrb<c5C«»Mle epmartmantm faperpattuuUnQ O2i0fntbu9 onere: in tnanmia roperparttetiUnbaa funt collocat quooenptopotirum. 15 rSismatcflaron autbieoiapcntcmuIhmconronanrianicopoDcrcpotdl. (fiacc ptopontt onae otareiraron coronanrtas aut Ouae Mapente conronanttasnmliicoaAarvpoire cofonanriS.tlam et ciatcffaron tolapfte no in mnlfipliriboa fer fucparncnUribiwfantconfltmre et per ptiinSpcrirfonfqae trerHaUta»tnferoallfiptfpomocft:eaquocpeilnfoniarfonfi.atBfcjrii ptimiroo fimillatntcrttaHa «5 niuItiplUiatnccvniultipktneicrupparttcuUreCTeamtnicmailum. quarenceptUotllfonitnmultlpIfritneqjinfuppBrficalarigfiemftunt-cmnwafitconfonirtaautio fnpparficulari autlnniaUiplid ronecollotaoa en.lTecnIconfonanrtenoicbocuiUKo 'i^ftbagouci feqn^tcsautotitatfifufcfplmosvtcrain. ergo rue cofonanric&iarrfraron anttmei^apentemuUam efft.-lef cofonanru.ct no moro irwfi firrf^er tquotquot ofonanrt cbiateirarS mImettfum copuletur crquotqaotriapcte;Dumvn^conron3tiame][qumtaptimil7uiBeefftcerepaIebfit. rSDinncto OD cofonantiain oiapcntc tono: nulla parabitur confonantia. iwji ijccp ao Diatcfiaron mfcmitonio. jglril I[Hqmerogptta IyhmavtfUS^^ 'S^malot geroaAiw iCiJp]©; «.Tonus Ili^l f794^^gefqBlconas (%! >')8! ilDUpente |IHl 4’0 (fcuttdftrofl II1 1 1I/II 1431 Semiwtmfimiaos I I1Ii)b|»! It i8ItoBW C-StnblcronotueongrcfrusnSrum eSfonanttaflneapbonfi tamen rnofid reputant melo, moMtla* mlbofqjaptumifenScp q*ffj impicatur roflb* nofhi nonf nfiarpanr^ quatmot tonis atqrma oteft boeeftfcmitoniommoK conilat.qniq^nSoum tofonanafit patcf.BcaptoeBJstbtettiailfletlHna riumtinminios ff5numero* eSfonanne rupentc.t crnouenanfi at<p octojurtamulmo*numeros roui.^ rucoc (natveniarrfcptemfHptaoismtuctotnbt wnlateoecemofer^jertertiam qulRtt Sritbmeticcs sare connet fefonaltert trrfquioftaufitquarcr ar ecStinet rtapente atepartanetS tonum.at manifeihi cfl oar ci7fdltca ar is no cficmulrtpletjiam reptem rttstim ni continj‘r bis autcertioautPrfccpsferccim.neqir r«perpartifulart8Cftarc5nam rnanumenrar ecOvnrecim qninnmcru8fummejiJp3r0nnllaen.rranrrfrttfBlimrenariD8fc»<narfirttntri6.iffitor_anfittu8 ar eonfonatiam rfapenrc fono6:nuHamparitc5fonantl3.et (ImlU argumento ariettoarcofonantiS riatetTarenrefqnitonotnuUafir cofonontiaw et fetfira figuratione perfacilepattrepoteu. fittame eupbona vocum congreino: qol irem fertam nficupant: fer quea ptima cwmwtoi tota rtflet apo* lomc.cftrrgobeenilnoJiiHaveromaiotrfonllatenim primavtricramlaineinqaattuot tonis et vna eieftfccunravero tribu* tonisaroabn* rtcflbU8.‘pzim5foBet1fb8rbppatebTpatonaomefcn: fccOram vero qpecontractio: cfl: fona-^ brpate brptton ticb««on mefomque autembTpaK.fl*'« pwbypatctquilidjaniietqueraefericfturtfcqujslibcrredarabu. ,8 C^0paclo Diapafon confpnsintta: in cbo:03 Tit aoiunscnpa. ISIC1BICgfonanruwapafott pfonSriamtnrt in libio ptobleumatfi lefhuuscft Sflftottks:eleg3n(Tlmapolcbcrrtmtep cft Cbojramergoabfecopermerlum pernoti c. et qoiaababcbert rapUinterualUbabiroro. ergo per riifininonem ab ar ebconfonat riapafon. C£oftfofKiiwaoi3pafon'4n fer tonis nuniincconfifti!. fco QmncpamplioJ: fervero iv toma:c^onat contractio:. CTBam per Vicdlma prtmt qoliKp wpl‘o ‘n»njaIlo CMMfisum.ft per nediit^tinmi emeretcotuonen in»i<>w«WK> POpUcl mtetiiaUo niaounr.erg0rent mmo re* fntit DtapaToa c£r«iiMXU|ct fcc esemfutu «inpUMe*. wnronat ergo maparon quintp tonis am p^feoafcF eaM»tno9BUb4mrmfeno:. Cl^apaTon :er otateffaron et oiapente confonantus comungitur. lo CAmiper PcdaaiqiUTTaw boius putrifaron et ptopeste in nupimitriicpartlculfiribDe funr cot loc*te.«tp«r tKPcaml pnmt Mptet mtcrDaLA et puobna mtttinis foppartloiUrtbus cfilongtrur. et PPiHn ntemalUuonfoninettspsfon tnteruallfi en(ht.igitBr confonanni PiopaTon: PlatefTarf (t Pispestc c^oBsatu ftnul contuoffust^uop cfVpu>poftttiin. Cf^onantiaoiapafon:qum($toni0etDuobadrcmironUsmino:ibu0 quetonumi ti nunccoplaii:pcrfiamr. ^nDequoq5mamfcflu dTcpotcfl; cofonannom oiapafon r<^o afec commateoiltanr. pteecp^tan ent PiatetTaron et ptapente cpitronSriant Pupafon tungunt. PiatefTaron aQt per fettil^MPOPbnstpmactrrinitpnloimiioteconCaretnonraraelbetPiapcnreenonatribnotonte rcmitonlo<9minoic.at puo tont et remtiontfi minus itres tont et mPemfemitonifi minus flmnleon lUnupuncp etfiafinir toni ttq» Puo rcmiiMiM min«2s.atqut pdo femironta muiota tonum non pei^^ rtuntnvrft sb eo pefteiimrcS(nate.tginir eonfonins Ptspafon qummtoms er PuobosCemitonits tni nonbusuioc tmifi muumelptcDt qu^sptnopfi tsm ptopoRtfi cftpertaF. CCotreUrlfi exPetnondra tio«tscaUcn«tie0epoteft.£tquob4o<tcf«eaeencinncruontu(lcac5maperuefhgare*Ramineo •ptlBOpio eoemtens canunsttr^ frt ronts et ab eoPjnmn tniiio ap metii cbotpe noram intenfa mapafoae^roMntiaqDoP tmer mePii nerul notam et vttimfifc}; tonot& Rgnuin cotinetur: et ptecc-- pennaeotnkrtoerit eomaiu interftitifi. CBonpta cr oiapafon cSfonanna oiapente: rdmijtur oiatclT3roii.et er mocoepta eofonanaa wateffaron: relinquitur oiapente. oempno autem er ca oiapente cttono rdmquinir fcfquitonuc. in'lP-’ln'a p»reet(ecun9a narim prrpenulnmj fognlrefonr. stem tperpcepenrc. Wam pcrpJeec: penti PiapaTon qutqi ronis tpucbusrcnnieouaminonb^conriatta quibus fi trea tones ctrtiQ Taiu tonifi mtnns tollas queper pcciml huius piapente cofonantia conrintt:rcIinquaturPuo rc t» tfcmi fomn mjnus:^ per fejai huius Ptaicffaron coronantia clfictunr.Pcpta igitur et Mapafon confonaeia PCapCTtr:retinqttifPi8tcfr8ron:quoPcHpnmfi.SeainPacaPcfanlitafcPeclaratur.TRQm ctquinm tonis et puobos fetnitoniis minotib^ fi Puos tonos tfemttonifi minus tollas: reimqufitur tres rom et fcmitenifi minus. lertifi conllmiUter.Pmpti8 ent aconfonStwwapenteboc cft aquin® tonis et puobua femitoniis mtnonbus pemptis mqua quartuot tonis «remitomo mtote reliquus riltonna vnnsirctnitontfiintnus.ftquorPiefesquotapotonus quot^iijcomataconttBeatPtapmbntPe pjehefioniafimt^lltmertn nuUa tamen equaUtcr:totatpriPiulontenqu3ooqaiPem piapafon In niulripUei ratione confilht que«nnino in quori^ct equas ptopeitiones quemultiplices n5 fintper frtagtllmam noni anthnentes PioiPi nonpotcit. Cftollii fimplcic cofonanria ;mouo cqnaliaictrto/coftimtoq! nScro oraifibilB cll. CSiinpUcescenfonfttasvocam08Piatenaron,Piapente:piaparon.peptarfn8ronaur2Tpiaomfe coRitat qet fneparrtcuUnbus interuallw rur^n^pwqomtSpnmt nuUoJpacio hunc In nXum pi!liBtpoBMW.ocPtapafonveroeoronanftatpeiubitmPlttfi.Vlamqoonamlntmieiusnammfum oaoeiVBiamtPooquaPtatusnodhlgiturpercotrelanfitncefimefecfiPefecnnptbuinscofoiiaBfia PtapafoBqconfilht tn.pporttoncpuoni aowinwnMnimelnPooeqaaliapartlef.ercopeauoffimrf neqt eapleonfooltia in ptors onobus pulie^/Ptrimetunb equaliawgtjpjofccfi, reUebocoarto <-«» ronjnrtioiwafoo te plura eqoaPtPBoerc.eft in geomemos Pianietrfi colle quaPtari wUccomcnfu rarcaep tp ruiiafi mnltcnni no ctt It 15 Piapafon acoteteffaron: cemfonantiam non efle. amcnaqicogreriow queaomopum cum pol^frfqutfonua^icorcotTOicucmtPtaparoflacotatefraronpfonantiaPianierm.ga.^ bnunua t<pte^.tiununirc5nomerf (ooronanePtapafonetcpqnatrneif triamtoiml loetipemco per Krtuai qniim Bntbmencca eap fconiinct oupU «fefquuerttaiquarepiapafon acPtaiefiard, III rt»eoa<w8rin0nencftmnltvl«a»fttrn«rta«,»^fBCpimM;uUn*»«lH»ceniie*rctutftitKr btoorifi qui remarfl nfi pars dl (eo psrtes. eSeal «aoasnus ac mvanfi popiar rabp^rQ^. Umidl t^Uur Pia^fon ac DtatefTaron proidR8.oIs eniconfonina aut in fepparrufirt ;aot t» imib dplid genere e^ Pl^nttione in factk cojpiofa pptdl ex omk» aoietmsti Itbwf tbagotictscofenfltflrtfleielesJcfilpOtttrttcnriujnbisPiapente autbtaptareffaroortpw yftwSna potefbrt bts Dtapafon coaptari fddJboc Inqmt ipco eO q* pup^e confoai tia pof^ tJi ptopomone rcfqualrera rll otatetTaron )bo In Tefqtuftia.q' fi puo fefquaUcrt aut r<fqmteTT4nutnert cmneMfpo nltarrptremi nuUatninoicen pt^omoembabd>&t.netp cn truppartiotlares netp potertit^t Piapafon ^cin2tU qA In Puplart pzopomone coA<lit:bac geinuta:quapmpUiM loaucm ertront renebuc'babd:>fit9g)Mmoem.3iPce ergo quo pacto anAoteles confonaoarft .ppomone* folas fucporricnlares ant nmuipUees ciFtcicn'ucpa rn^es ^pwtfos nnlie Ant repeptans. £t re 10tl?olo<net cfi ‘Ibrd^agonas magia in noic 9mreipfa Ptlicnno putaapa tBJeo bei^ta baomus. CBiapofor} acotspmtctin tnpUci condUit ratione, dk^ oiapaTon ac Mapente^n^ fonannavna. C^XI enim Piapafon ac otapente in tripla rSnecofUlatxpoc iPeo eft ep PuoPeciraa pnini ey Popbct atqifefquaUointeniallotrtplet:nafdfEnterualLQ.PHplet:a6tf fefqnaltep font ofonananPiaptfoa etPiapenteiafualU.i^turiDncreoronatiePiapafonacPiapentefntnplardnec^inttnr.rePcfifeniU iam fatis (U exploucfi bfic cocentfi mooulate;fuauiter<v ao auoitfi puenire: ergo per E^lftittn^ is concitus ci>nr<^naeft.qaoPefltotum ptopofitom. Cgtapafon oiapente ac tonug:mcloo atra confonantiam diauni. I17 I!!*^1 1~c iI>fI JSSSSoiiltooejnna fttr airts ou. f•» fmpUi. fi«trwam&i «uoM .«on «SXSfcnmratia pCTceptum ert.ft BPifininoem erit pfc^a. qP er«ini^pfl. o^lAxKtSftao&rocCTmamoPfi in ternuniaquapritplearip uilw ^onine F.‘£?LSn?Sf.p“wV»»«»Up«.tc,ni.p.=nS.frfcu^ :ffircSSrnSmScto:oafuoo:Dmcrab.us«^^ fcnTuixrccpn^ bilitcr ctpcnri. h CSitabpata cl?on)am qua mot«i^ocH‘««fi®®'^’®*®*®®®'t^,'‘^®f?|,*:M^^aron.abtobperoctaoaPupente8bteb «inbffl«f«ii/ipto«ffmtmm'*bjintceCT«irttiJ*fnt<Hiu*nnim»iJpafort mm c3<fni«wRficrt fmirf inwftatnSe f»ctfi ri qtS md>t«n«tico gftfjjponcbaf an» ^poiUQ. CiPcmun cnarmowco genere reguUnter oftnipgrc. 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