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a partialẽ eundẽ ſcʒ aut multiplicẽ ad b in b det ꝓuenlat rlꝛ qu

ad b ꝑtiale. Deiñ duco c vnitatẽ in b det veniat lꝛ g qͥ qꝛ vurgaem anen p 3 primi eemoli eſt eãlis b d:ſed b d ſupat b ſola vnitate.crgo l⁊ gſuperat b ſola vnitate. Hanc. 7 ducak llu oug⸗ aggregatꝰ fh eſt młtiplex ad b:ſed aggregatꝰ fh ſuperat totũ fg ſola vnitateh 81 Ddeuy alith gti ctu a partialis ⁊ c in b d equat᷑ e q ſit ex ductu a totalis in b d per decimã primi 8. ed cũfg fat erd pliceʒ ad aꝛ f̃h muttipüee ad b. igik᷑ e multiplex ad a equalis quidẽ fᷣg ſupat 5 vth eſt e eẽ mul vnitate qð erat demõſtrandũ et propoſitũ quo advnã partẽ. Sed rurſum demöſtrandee ad bſch multiplicè ad b qͥ ſolã hʒ vnitatè ſuper multiplicẽ ad a: aufero vnitatẽ ab b totali q ſi eſt dabileeſ tiale in totalẽ a et proueniat e manifeſtũ eſt e eẽ multiplicẽ ad a.et duco iterũ b q 45 det ducohpan equali aut multiplice totius bet proueniat f manifeſtũ eſt f g eſſe multiplicẽ duale inapartial b din cvnitatẽ ⁊ fiat g h qui qꝛ quicũq; numerꝰ in vnitatẽ ducat᷑ ſeipſum pducitgy dld dehnde ducs quare totus f h eſt młtiplex ad b. Sed cũ per nonã pꝛimi qð fit exb ꝑtiali in ac equet tequalis toti- partiali in a et in c:⁊ ille qͥ fit ex b ꝑtiali in a:et b totali in c ſolãvnitatẽ ſuperaddat rf qäterenet in a et in c. ſeqtur ergo ꝙ multiplex ad b ſcz f h ͤ ſit exb partiali in aet b Dtiali in c: wic texb ptil ſuper e multiplicẽ ad a qui ſit ex b partiali in a c:quod eſt ſecundũ et totum pꝛop ofit 3 addet vnitats

¶ Pꝛopoſitis duobus numeris cõtra ſe pꝛimis:mulnplicẽ cuiuſii inateſh — IleI EDard uſlib cuius ad muluplicẽ alterius differẽtia ſit dato numero equalis. go nuchan

dꝛĩa g ad f eſt nũerus datus c. et cñ a ſit pars d er dps frr ſilfbparse tepan

4

deinde ꝓcedere vt dictũ eſt.Ceterũ ſi minoꝛ quoties põt de maioꝛe detractꝰnãᷣ ſalãxv⸗

ſed vlterioꝛ fiat ſubſtractio quo minꝰ per pᷣcedentẽ deran poſſ renpnrolarelihan a b nũeris contra ſe pᷣmis et c numero cut ſit capiẽda dꝛĩa multipliciũ equalis et detractob aba 3 ties põt remaneat d et d detracto ab b quoties pᷣt reliquꝰ ſit e:⁊ e detracto ab d quoties põt deau⸗ et qꝛ tandè per decimãquintã huius relinq̃tur vnitas ſit ergo vt detracto fabe quoties poterit rin⸗ quat᷑ vnitas.ſi ergo multiplex a debet addere ſupꝛa multiplicẽ b ſumo alterũ extremon quielt inſin ĩpari ab a vt fet ſumat᷑ h multiplex eius qͥ addat vnũ ſi uper multiplicẽ e ⁊ ĩ quo ſit fiᷣmg ducatyi det fiat hlꝛ qͥ addet ſuper multiplicẽ e vnitatẽ:qm̃ g in d qᷣtum in e vel eius multiplice tinf. dengt ergo vno de h lꝛ remaneat in lꝛ vt ſit m et vnitas equale h ſitqʒ e in m la fml qͥ ductus inb faciatmai addito vno fiat o quẽ numerabit d eadẽ rõne ⁊ ſit vt ſcᷣm p dͥ ductus in a faciat r: qui ſili additlina malripliee b qui ſit ꝗ vnitatẽ. multiplico igit᷑ r et q per c numen datũ et pꝛoducãtur s ⁊tqͥ eruntmu tiplices b ⁊ a int᷑ ſe ð˖ĩaʒ ſᷣm c numeꝝ aſſignatũ ſ eruãtes. Tertij elemẽtoꝝ arithmetices Joꝛdaniſins

I fuerint quotlibet numeri continue pꝛopoꝛtionales: protin ales: duo et duo pꝛoxm erunt commenſurabiles. 3 h

¶ Sint ab cd quotlibet nſieri continue pꝛopoꝛtiõales:dico a b eſſe cõmenſurabiles. Jtẽ abe uen Si enĩ a nũeret b manifeſtũ eſt a et b eſſe cõmenſ urabiles. et qꝛ q̃ pꝛopoꝛtio a ad b ea 5 adccdd ideo b nũerabit c:et c nũerabit d.quare cõſtabit pꝛopoſitũꝛ a bꝛb c:c dꝛeſſe cõmenſurabiles atllon nũerat b:aꝛ ĩpoſſibile eſt tũc a et b eẽ in ſua pꝛopoꝛtione mĩmos.nã ſi a et b eſſent in ſua ppoꝛtinnen nimi eſſent ꝑ decimãoctauã tertij adinuicẽ pꝛimi:et per decimãnonaz numerarẽt quoſlibet inſua poꝛtione: ⁊ qꝛ q̊ pꝛopoꝛtio a ad bꝛea eſt b adc.a ergo nũeraret b ᷣmus cõtra ſe pꝛimũ qs eſtipoſſti et etiã oppoſitũ poſiti.nõ erunt ergo a et b in ſua ꝓpoꝛtione mimi. capiã ergo ꝑ viceſimateriisz un e f mĩmos in illa ꝓpoꝛtione qͥ per decimãnonã eiuſdẽ numerabũt a bim eunde numeri cquatufn zet conſifr bc et c d. ſunt igit᷑ a bꝛb c:⁊ cd adinuicẽ cõmenſurabiles: quod eſt pꝛopoſttũ

CSi nũeroꝝ cõtinue ꝓpoꝛtionaliũ duo extremi fuerint cõicantes ent ninscn oes numeras maximuſq; qͥ oẽs numerabit erit maximus extremos numais fr oc cöſtat ꝙ ſi aliquot ꝓpoꝛtionales ine duos cõicantes ceciderit: totidẽ inter ticʒeat et maximũ numcrũ eos cõiter numerantẽ cadere neceſſe eſt. Palãq; ſit enãmaunds qui binos ⁊ binos numeroꝝ pꝛopoꝛtionaliũ cõiter numerãt eſſe counue ppouiila

¶ Sint ab ednũeri ↄtinue ꝓpoꝛtõales quoꝝ a d extremi ſint cõicãtes:dico pᷣmo aliqut eſſeruna eos oẽs cõiter nſerantẽ. ſcðᷣo maximũ nůeꝝ oẽs numerãtẽ eſſe maximũ numerants extremos. i ꝑ pᷣcedentẽ a b cõicent ⁊ ſilr b cetc d:capio ꝑ 1s tertij maximũ numenꝝ numerãtẽ a baͥſit eꝛrman nũerantẽ bc q̃ ſit f.et maximũ nũerãtẽ cd qͥ ſit g.ponoq; hlꝛ mĩmos in ꝓpoꝛtiõe a ad bet perꝛ ſa et 19 eiuſdẽ efg numerabũt a b cſm h:et g nũerabit d tᷣm lꝛ.ſumo itẽ maximũ numerantee f die maximũ numerantẽt g qſit m:per vicelimãtertiam tertij inũerabit e xmfim heet itr lnn

mk ⁊m numerũ g ſcðm k. per. i.tertij et l⁊ m ſunt diuerſi nuũeri ab h k. nã ſi ide ellent hinſenat ceret e ⁊ u in ſe pꝛodnceret g ⁊ h in e pꝛoducit a et lꝛ in g pꝛoducit d ⁊h la per. ls.tertijſunt 12 mall pꝛimi. erũt ergo a d per duodecimã tertii cõtra ſe pꝛimi qð eſt contra hypotheſim· ſunt erse Won ab h i:. q ergo h lꝛ numerãt l et m equalit: numerẽt ergo ipſos ſcðʒ n q peraʒ:terti eſt morim