Tractatus proportionum

'UtL J, ..t '' 'f' cAtu-bL-y. 3-‘d Cda/'* ) r r \ Ipojrionu?; editus aipetroarpinoraartiuiniua igiftroj fimulciiin oiiabiis queftionibus pbilofopbie quarumvnaacttonem veftigat ap:opojrioue inaiojis iiicquali tatis. jeunda { vcroqueltio j ucmDcantipart' jllafuvbi ferine totum fim Idamentummetixotojumcontfnc tur.^ jaifum-ioepxceptoptelariim p2eflum.3nno.2^.©J£X^. iDctruuafpinoia artium magiitcr fuo oomina di^NUchacIichacon^Iari(Tiaiorectori5alinan.acadcm!e.S.P.irnpa:tic. iD^itabam masuificenturime iRccto:qitod* dam cifc genushominutnmaximelimiliiim torpcfccncibus anlmalil us* qu«uclml mortua in tauernis terra degunedn Ibporc hlcmcra tota ttifi gcntla. Homines hi iunt quorum cft tanta pigritia feribendi qbene co gitaructutminlmuiaborcm praferattantoftuctui quantus iperatur cx docuinls eorum bene peniatis dJgellisati^ tato labore cxamlnads. Ho rumnumerofui aliquandoalcriplusparceadi labori nominettumedan) neuidctcriaccabundusd^ambitiorusmehumlliorem meisdomtniscor regentibus pnerrLquod abiit.il!i5.enimm(X'emgercre;graruiii eiTecm iiious in rebus dcitoaui.Iam ueioablUolomnoexpergefaaus periricui ftontenutanqlacetiensfociosmcosniinimo opu&ulo propordonuniii ** ut ipi deabditisarcanls kpenctralibusartium'^ phiiolbpbia; xdantafi quid In lucem.quodfi fecerint ad maioranollrse phttofophJ» qfcp«triulmus dc coglcauimusprouchar. Honutus flimctiammagnificcntifiimcieccor q> tuopa^oclnio libellus nen poterat noneiTetuiustcome datus:at^.uclUfl1n)Us:ut<juilblcsettampDgnantuconctIlare:benandapaccare:fluctuantemireinpubljC3 hinc.indec6inotani»pefiinilsncgociacoribusintranquilicatecdltjcuere.Quado vn^cathedramifiefunc fineftrepiiuuocum:negoclo:^:at9fineajppcIiationibusad curiam imperatoriam nifi apud te. TuaJpi^f^ tuna oiaruniproQxrafaxitdeusteincoiumenreddercutquaiiscsiialemfacias rempublicamtua» Texe rem hic tuarum uututumfuiiorecacalogumnifiteeumagerequifdam.immoderadoriste laudis concem pcoccm&efiebonusquidcrt mauis.Petcitus edafui huicopufculolmpriai^opftc cibus mecrutn fcoU idcoruouquibus non medebere ^eor§uere£llitsct&actibus:quorumpr£ecib9 mlhiuinimeftob ccperarc. Accipe%itur iUuIitiifime rector has lucubraclunculasmeas prlmidas fapientix mea; queeun^ Sila fir*Ncc munuiculi patuuaccmnegligas. £fi enim doarinamai^mecenifi outium prxfagki^ nuilca inalotatat^quericanstquoanimoabliudendbusrccipiarurrfpcramus bcniuolothilaritquodcura&crie magnitudinealtcwm hiiius brcuiiatem penfabimus. Artaxerxes enimperfafuro rex non minusregtum ix humanumexifiimaDsparualibencer promptc^accipere ^magnatribueretcumfiblin uia ob equican* ti paupetquldamae operarius homonihilaliud habensiambabus aquam manibus cproximo flumine haultam obtulitTct io^ndcfufcc^^ arrifitmon leiquc donabiuui utilicateiied promptaeius qui largie bacur uoluntaiegradaro medtus*Vale fccUx ftudioforummocccn^^tuum «SpiuoiitmQumetotuoiuQi afcrlbcnoftmni^opeUam boniconCule. fratcnRodericus coxucra ino nachusAdkccoiem.decaftichon.:» .Ythagoras cdfust{lagirita.&|turbaPiatootsx EtfcurinacohorsttcnumeratcdoceC. i;i luniot his magnuslimphis {ordanusamidss Cufaquo^infignistdoaus&ipfcfabcr* GlariusaftparuolepidusSplnola libello. CAglaophcmusadelLalterApollo rubet.^ Arduacunccaipquz fitpropottio retu: Oftcnditrithxnispromptusin arte fufsi Lucida roens numerosprifeo^Ccandide lector^ £xcoIuiucaniosIpicpacresfequicoi;« Cractatusi g55tracfaf«sp2opo:n'onu3 paarcutifTimamacegregium ooimnnm iptcmini a ^r:floranunccompotiQ}s:omnct<]}0(ff:cUirfltcscon (inens. i/> cil Duarumquatita- \Bjmmcoquodwri lO- 1^carumconunuK cofababitiidOiDici ii^cequccfiimi ^ItmiiotKopomonu jdrIMiiiljad nunus tnbua termtnie conitar, ll^cr quantitares mtdlige vututts amnas cpatTuiae^ncaifionesrr rcmifTionca:'! veio citatee. rafia piura cr<^parab)ha.)7fCitur meo quod vmcpcanan conuenitper quod intdligaa genua fubaternum-^Warn ImeanS babetalicpiam pzofVMtiones ad caloKiTKnec aotmcrro.Stmilittrquanctas ouirre ^adeoneIhaamra]{^ambabetp^opomonem:qula t'o Ium eonumtunt tn genere gcneralifhmo. Dicitur eo-- ca babitudoquodbupbaterinteQigaar^no modoeer quia mtinitiad tinitum nuiUetlpjopoaio* lanam miinittadintinitumimBap;<:^cmoeft> 2ibo modocata ideilDaerminata babitudo Tinditullbi' lia :vtpoce.4>ad.2.piopomoconti!biem indtuitibili- TRamft quamummaparum accreicat nnmerae qua* temanuavd0!m.nuaturbinanu8iam variata eft^)- posto.tJiwmtamen nonfacitue pxtenbo per qua nrates intelligo «^atitatem molsipei fertiemia: que emmfunc perfena conibemmus m^na vccaremc' C3'beaaimi!ugummtm:inb:8quencn mdemagna fune idemeft magnum7perii:ctumeclibK> De quanti* eatcammcmicoq^pafettloeficntialts bommispotcft Dici in ouplopcrftctioi quam eaqueequi eftma: valet ergotHioeqoi elTenttam perfecti vt vnus bomo cum tnio equi nonbabentdc pa^cnbneeffentiali pius qua vnus equus iurta 'iPoqjbwapifiilo De fpecicpaitict* patibnelpccid iMurestvminesiunt vnusbomo.^i enim DUOavii cnaicpitootcs ctfmtialiter quam .vn^ cfTcnt dtenua fpeaa Duoquamvnns quod crt falfum tViive mdcmibi fuadeoq>vna ipedes iit in intinitum periectic^ ala. ^^,^«j0ncvideanturdrta matenam p^5poT tfoni8Oifficufcaft0abque.SitpnmaDif b'cultasqneentirasntp»:>pomo:adqua refpondeoq^aiituten^umei^fmno' minaks;Tfmrcak6.^cundumnomi -‘i^^^^^^itafegpiopomocftiprcmctres p;opo^ qbnate.^ioerfabferirtam pjopofitionem concefiit jCaknlatcum capite Detntenftone ^pomomaions mequa]itar!'deflp:opombmmi>:is inequalitatis. Sed a^dredcspTCpe^bptofubftratoeflres pKpomo-- nare fcd p»per fc tignificato c(pedamivlatio.'fi)abardinns tamen p7imox.fua.pw.rcnet q’p2opomog quantitas continua cuius raao ert ilfa.COmnequcd cftDunfitMkirtfemperauifitnbaeflquflntitas cenn* noaiomnis ptopomoeft Dmitibibsui femper Ditulibi ipjopojtiomiin. fo.i/, fioiergo omnis pfopoftio en qiianncascoiitinua:i1Da io; patetper Diiterennamipam itriiUtfignat tertio pbificomm inter quantitatem coimnimmi Diteraam Iflam Dxit q» quantitas continua mDiuitioiie non ba betnotum.Dircretaveroquantasinappoiiaonc 110 babet ftotum^IDino; P^^batnru^uia ptoposio Dupla Diuiditur mDUOS maitaates Dupic 7cv^ttuo; quar* tas DuplCT fic ti ne natu. F:::^^^^itclarataDiffin:ticncp:opomcnis re« tW^ fl Diuifte nes ofTign^c.CiPn nr|^^«^matnuiliopxipozciciiLm(pedamcq Iitatis:quedam inequaiit^isiuiecpalt (Itatis maicns incqiialuacist^muiona 1—.. MI. J;ncqualitatis» Ciibtopomoequabra tiscil queeft inter equalia.'}lnc(paluatis qncctt inter uicquaba:maio.'is incqualitatisqueen maiozis ad mi misnnmons inequabeatts qtte eit minoiisad maius* £panplumpnmi.4.ad.a.£i:emptum rcoindi.Duoms ad.4. €&tneta $pfopcntio tnaiovs inequaiitatis octoadquactuojDtcitur Dupla»7 minotis me^iab' t3tis.4.^S.Diciturfub^ipla. CDiuitio alia. B'tfloposionum3io;ts inequaltatis qiiftt tuptet en genus vidckcet i]iulQpiee>iuper Iparticalansifuperpaitiaisimuitiplct lup jpaiticulansimalnpiet luperpamcns. ICifbzopomomulai^drinqua maiot numcruscontinenninoKmpbiriesade<)uatc> Dicte feries quiapK^xxtiotnum ad Duonon en muitipl^ quianonronnnettem^iusbmarmm pturiesded fCr md.Dicitur adequate pioptcr epod pf^jcntio quin(s adouononenmulUf4ec:quialtca contineat phirtes: qmabismon tameadeepateiqnia adbuc fuperen vna vnita8.1^deptopottio multipki: quecontinet bis ta tumoiatur Dapla.7 queter cantum tnpla.7 que qua. ter cantus quadrupia.t lic tine natu De quo fadesdta taimlam. 234567S9 t I tt1t1I C^nqua notabis q’nuTnaifuperions adinfdfoami pmnumcnf^S!^a:7 fuperion^fecundi ^mldioreni feomiiimeft mp!a;7 infdioasfeciidi ad fupeno rem fecundumcH futKiipIa,i:ic quo infertur in muld f^tdbusptopoaioiubos ett Dare minimam fcd non i Dare manmam.Dup(aenim en minima 7pon lUam non Datur marima. ClPtopoaioiuperpaitiadaris en p^^nio in qua tnaio; numeroscontmet mmoxmlemdtcumbocid pcrenpars alupota minonsivtnmner^ remanus co* cinec binarium femel xfuperen vnicas que eilpars ali. quota ipius buiaruaponiamemedietas.l^ndeparff ^(pocaenqueabquoticsiumptareditipfum totum vt binanns refpectu qnaRmaniapi bis (umptus ^ «quattmanumxbabetmulcasipeciesvidcbcet (ee- (iuiatccram-fctr(pitemam.rei:(piquarcain.7fic fine na Cu.sE>crquiafccra vt Dicii •;?acobo8 faber non bene Did CirrfcdDeberetDiafcqaialtCTa, sxquialtCTa Dicitur a fcquiqoodentotnime aicera.mtdi(tas,0e()uiteTtias f^i(jnod dlcotunuT terna parn:.Ouandoigiturc6 tuuentahqusiuimerHsnwotabquem minonm co a9 tractatus 'fcpcnun£v{mt!!i9qnattnoz tefid.tf>S.942.vbui2.a«I S. i.9.«.e,t.ppaaional«as gcom«nca.i.«i.ad.9.T«9 ai.i^e.ppoaionabtas an{bmcn<ra.7.i2.ad.8«T.S^ tfxilppDznoRalitasonnonuraqzquaitse ^pomo.iz« ^.(i4ilisecccctfii&.t!«ad.S>adejcc€fTtim.8.ac{.tf,vtro bicp eiiim cfi oufdapzopoaio. C!^otabisq’Da£»inenDocuinm^mrnpcnDead m tieniendum q^pomoHt xmiue ^pomonie ad aiora; boc modoquerere quot Rmt media geonKtrica mter oaremo» numeros maions^poaionis coparate vno addito vtfivts lareque Iit ^{watooetupte adou(^5 reipiceqiiotfnntmediageometnca inter.8.-T.i.in Pupia ppozCidne -r inuentib (p tunr.z.vc5^T>2,qz quatiseit ppomd.8-ad-4>talia £[l^ad»2.t«x.ad.i.t tic erant Duo media aeb^ vnum xerunt tria tfic Dices octuple ad oui^meffe ppoitioncm triplam tcaufa e q: moctuplacontinent u-es Dupk fed funt aiique ppo; tiones (]pe nonfequunfbancregulam bocetl boc documefttoncmpoffufTmsDirctmcreqppomo ticvni^ ad alteraiipt tu femabis boc Documetu mlUis in qb^ i^^muDocum^nopuaiitvideas quot funtDup:em iHis ti ecDoplis iflecompoiianf vel tnple fi etiniAis x <0^3 cmplasvdtn;^vm^ ad akeri^onpias rtnpias tccempiu. <c!i8fcire c^ue fit ppoztio ectu^ ad quadra piam tu vide extupla quot Duplas babeW xvides ba bcre crcsvc5.i.ad«2.T«2,ad,^.Xf4«ad.8.bocipmvide in qdru{4a quotbabeattmpiasT vides babae Duas ztnuDu{^a:padDuasDu]:4asefi^pomo fequioltcra XIkDices ooupk ad quadruplam effe j>pottior>em ie qwakeiiitn.'3tem Ii vdis fcire que fit ppoitio fedeeu- ^ad octi^Iamvide quotDUf^babe^ ledccuplax itemoctup^vides fcdecuf^m babere qttuo: xoctuplaintres.T qu^tuoiDopIapadtres Duplas efi^poi tiofequitcrria xita facies mommbus.Sedfi^pomo compararafuentcdpofkaei: Duplis xaltera videlieet mino:ad quam tit comparcoej: tnplis vm^’ ad alterameti ;ppoaiotrraticmalis oposetemm^arru^ Du piis vel triplis coinponi.lDec regula fup»i Dicta maet ine valet in, multipicibus DummodomlUis lit aliqua I>?omo rmiltiplep. lftam malgs lempacfl ptopoaio irrationalis quare feq^iaitat adfequitertiam irratio* naliseit^poitioOtcmfequtqaintead fequi^to-J eft irrationalis piopoztto. CUndeama Diffiedtas. alique.ppomones fiiperpartiailares aut VA*:uperp3itiemcs;aiitvnafupcrparti£iiIariST at tcra riipcrpartiens noncccedcnto numerum binariuj potfinc componeretriplani autquadruplam autmaio rem bi8.Hdquam DiMcuitatcin refpondkop^^Tcnes ‘lp>nma lit-TflnUe DuefiipcrpaiticularespofTuntcom poneretnpiam pbat .ppomotripla conflatet vna du piat vnafei$n'a!teraied luiUa ruperpaiticulans pdc equari Dupleigitur mtcntu.0ecuda ppo.qfluIle Dac.p pomones qua? vnaefl rupcrpaiticnlWis xakera fup partiens pc^mitcomponerefriplampateriqznnllam perpaitiens potDuplc cquan-^Tertia ppo-Blique Duc lupa-partiaitespofTuntcomponere trifJam tmaiorcme3.Sccmplum.0upcr Rtpartiens qi^as tliip quintipartiens fertas compomintppozriooq cft.77. iid.i4.queefttnpl3fnperquinnEncns vtgefimasqr. tas.tIta DtKfiipapar^componunt vnasmplaniimo tnaioKincaiec quo infero quoddarutmk conc^anu»» t{>:op03t(ctiui!i: l£2uod licet reiuicansq: Duabus icquiakaris cqualiter qicedat ppottionem DUf^am fient ppojtio rcfultas p: tribus rei;^ialtcr:s ccccditcnplamitnppomo rclnkds eequattuoziequialtcris non cqualiter crccdit quadrupla (ed muko magistt iimiliter ^ponio ccpoftfa ep quinc^’qtiiniuplam uonexcedit equalitenimoer quattuo2feqtitakeris rcit^t^tponiomaio: quintiii^ tcp quiitc^ feqaltcns refuirat ^pom'o maiotfpcut^a, imd maiOirepnq}lapmnp>i^amvtrobicpcile]ccefi^ fi ponio reqitio(tacia.reaidum p5>qtppcnRo.Si.ad>i«.q Cii matozquintuplacomponifexifiis quattiiot.8i,aa 54.M>ad,t($.;<$.ad.24>24*ad.iti4)ua?(9)eIibd€fifequialcaa-Similrrer ppomo»24^<aii^^queefi mato; fertupiadmomaio: fe^plarefuitat er iftis cftincB vc$ 24;.^itfz.id2«ad.i0g408,ad 72«72.ad 48,48.^^ HOdeadmaiOKTnDocfnnamiam Dictopqji»- poSiomaiozis inequalitatisminoris in^ualu Catis tequalitaris fm Doonnam )&;abardini bme» c.ftia»ppo:.3f(p fmDoctnna jCaIawn.f4'mo Itn ope non funt adituitcem comparabiles ita q? fi petier^ anfubduplafiemaioiaut mmo: quam centuptaauc equalis tu Dice5 petitionem clTe ma^l^ampxdicte^p pomcrnra nem funt comparabiles tfta vnaxa? nb altera maio;ne(^mino:ne<^ equalis ficut^ifi-bmo oeedoDidtq> wuminfinitum 'non cll akmcompara bilcita q’ vnuminfinitu non efi altero maius ne<^ m£ nusinei^equale. «^^Odectiamfivisftirefn <jja.ppo:done feba- «/Xbetvnaquce^parsaliquotaadakavcfivfs icireinquappomcnetcbabecvnatettiaadvnam quaf tam multipicab! 5nume?aquo Dcncmmaftertia ponumepac^oDenommaf^iaitavcMccdotcr quaeo' runt.U4)uo babito imioiifii nmnenim babentem tem' am Xquatam-vidcergoque fit tertia Duodena^ te quatem^us.t que fite^arta «lodenarfi tefi* terbartus babetfe igif tertia ^i^rtam in fec^itertia. de^luaramtbomam pama parte p2Cf^onum.c» octauo fuppofitioneqoarta» tf^UfCUlO pwpoaionum iticf 'bi in mentem 5?vera amiacia non mfi cum fenore ptomifib abfotintfideo placattp:ofen(^vtfompar fitqbi^lcnbercDcBnn parilU^afii, ST '^^®*5ftnrvtnmiap»po»(onccqDalitafisvdminonBin^^ltatistiat aene fient fit a2>pcctide maioris mcqliiatis. Ybccificme qDeftiom'5 pono ali Jt^qoascondoliOTCs.lPnma efla pmporiibeminoris riicqualitotis titactiofifitfolacoi paratibigfadiri Wtn^^lasvt»d.agittcaIc«^vt.S» l fl>bifice, i^pjCrmatmaoercactioiictTcode inofitattioapio p^cnfcqlitatis.imoreimfliitit mgr^u tmmons arnuitans qua lU patTumr£iiften»e:agitui patTu; Pamodojdiiiuet 9maion quantifare fojme liibitanalis quccus tlU qdaiitacciup^rcfiftcnPatT lictM^taut» ’4x.Dctoqi>comedift btbtfrcmiflbm non agiem in tcnfumnt(! cu mairufacto ftxffTmuw.<>cgrotcratu. CScoa maions tncqlitatis cateij adShgidstatc n5 tit «tio oco no fufpcdenfe artiuitatg calone:! oiT?^ alo» gfguis Icclnlis. nP5fiH>pfitoq»q tiUKs contrarie fe ccpatianfad impkra larittidme m grsd!^^rcmiflis Iit vnaite calorebfis xttfafgi.vt.!. Tdiacalore tt.;.frigi;Vt,u^n3tir^<i:6a.b.b.agitm aialcfedcf^ tcon^opoido ftig^a ppoatce./.ad.a. ^aitripb letVJJalffra-CTgo.a.agrimfrigiditate.b.a p po:tic!!e.6.ad-i4ritrertupla.¥la titn^ feqoiakera pocintcorrupere qre no fctti?p}a.2Dicit bcticA q> no lemp a.pportioe maione mcqbt^^emtaetto naeft regaia ojgcr.gpotr^agitjptet afluntlare fifai rndccu^i^^mafifm^uno^ tit amdrR^Oc^, rcjV<itii,b.ir. calore Tnon ccotKrtb. CiTcmap^aiiqiwdoapporttoneinatonsincqlif^e non tit aeno qn fc«p^e£ incwtenife mna vd pafTom non b3cadere intra fpbcra actioitatie agcnrie n4lie>^ mSp3 oe aq inckpfcdra xt McicBrilbcn eti ciaafa gfu pen‘^70cl4>!devndicpcircudatoaertrarifrimogn5 ' t>c!ccndit neocfran’ranfTimottKcolonamtinita la ptdea qoitict ac£trovna kuca no beTred&qtno1:9 pn ctu medio q^tiatmmoliovntaerlivtbenebiirit tkc gvn‘^m fwno fniap t*ilt.44.q^4^.?. Safido f5 qr aiin^ pigcrit pdeae qe non leo gefl calidior afino 7be ferro ^diqt Aratio cameli7nonGeo gersAidior .q$ coenit er eanfa oicta-no cadere infra fpbal acriot^ t^8bnius venit er bocg> non1^ vnd qntate qeppe laeocubapaifuiapiuapatifabboctnbab boc COuarta0JLoqnertdo 6maiorAatet minontateqno ad actiuitatev rdttiMano titactioai^iortionccqlira tisnecmmoneincqlitatiefedregoa^ipqmone mato ns inequatit^e no tn femp f; moderttionib^ilbe »d iccneptT conduncKiee noArae» JDoerinequartaj^iOomS intellecti primopnciparrmllaf.^^oloq»in vna cglibra fttvn^ magneo vni^ibrein aba veroah^fantipcdenecuvnoferro po deno vni^librcno&is braebiopegtibx ad ahquaparte ocdinacergoa|>porti'o nemaiono ineqhratis no tit actiOfCticagg^ibitota ca eft g» ferrn no 1tiftincf aftateratioJafotomagnete 7fo tos magneo ibi pondcratfuptiatcra-vttticui&raoar' din^in p^opor-7 antbor oefer inccuentena:]$netettia. ^Siiciidobnnparrarguofiimendo vnfimagnete .C-^g fi poiTit aorabeicvml farfi muidaira m^neo q<i4 ferro in buao medietas ectitferpm^tia attra b;tur fs e.pportio coionctop toiuifopwgomm ffatmt^erpoccrattrabereT eratjportibminons m .cqfatatreicrgo a.pportice minoris ineqltfatis tif actio. ClProfcfatioearofinceoahq.lPrimSg licetvnn .facdlncSpofifi vndicpccfcmiccmagnettB frrrocriiec in medio pederanon occhnasad ab^apaxt&.S<6o noeo q» ti cet vnn fenti ad modoroee tigR Tcaic&ro magnes 00attrabcrctJad qr nonpoti!’ ad vna ptem qnladaliamXeittobeJmcipidi oico g> 7ti magnff nopoTTatrabere ferro alug? viif-^bK Diuilo ferro ioa as medietates potfetq: nbcfi tata retiAcnria in lUis o< mtisvtincbioncto. 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