Alvarus, Thomas
,
Liber de triplici motu
,
1509
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Secunde partis
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39
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duobꝰ numeris ſe habētibus in proportione ſex-
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quialtera ſubduplum maioris eſt ſubſexquiterti-
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um minoris. </
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<
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xml:space
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preserve
">Probatur prima pars / quia in caſu
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illius idē numerus habet duas proportiones ma
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ioris inequalitatis ad duos numeros minores
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īequales puta triplam ad ſuū ſubtriplum et qua-
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druplam ad ſuum ſubquadruplum / vt conſtat: igi
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tur proportio per quaꝫ quadrupla excedit triplã
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eſt proportio inter illos numeros minores puta
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ſubtriplum et ſubquadruplum / vt patet ex prece-
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denti: et proportio per quã quadrupla excedit tri-
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plam eſt ſexquitertia que eſt inter numerus deno
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minantes illas / vt patet ex concluſione: igitur in-
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ter illos duos numeros minores puta ſubtriplū
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et ſubquadrupluꝫ eſt proportio ſexquitertia / quod
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fuit probandum. </
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<
s
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N13BB8
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xml:space
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preserve
">Et eodem modo probabis reli-
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quas partes et infinita talia correlaria.
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xml:space
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correlar̄.</
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>
</
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<
s
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N13BC2
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xml:space
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preserve
">¶ Sequi-
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tur tertio / vniuerſaliter talis eſt proportio inter
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duas partes aliquotas inequales alicuius quan
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titatis: qualis eſt inter numeros a quibus deno-
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minantur tales partes aliquote: vt capta quarta
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alicuius et etiam tertia eiuſdem: dico / inter ter-
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tiam et quartam talis eſt proportio qualis eſt in-
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ter .4. et .3. puta ſexquitertia. </
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<
s
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N13BD3
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xml:space
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preserve
">Ad quod probanduꝫ
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peto primo / quelibet pars aliquota alicuius de
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nominatur a certo numero vt medietas a binario
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tertia a ternario: quarta a quaternario: quīta a
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quinario .etc̈. </
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<
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xml:space
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preserve
">Peto ſecundo / cuiuſlibet quanti-
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tatis ad quamlibet ſui partem aliquotam eſt pro
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portio mĺtiplex denominata a numero a quo de-
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nominatur talis pars aliquota: vt cuiuſlibet quã
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titatis ad ſuam quartam eſt proportio quadru-
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pla denominata a numero quaternario a quo
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denominatur quarta, et ad ſuam tertiã eſt tripla
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denominata a numero ternario a quo denomina
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tur tertia: et ſic cõſequenter. </
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<
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">Quibus baſibus ſup
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poſitis oſtenditur correlarium: et ſit a. vna quan-
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titas: et ſit h. vna pars eius aliquota: et c. alia mi-
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nor pars aliquota eiuſdem a. et ſit a. ad .c.f. ꝓpor-
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tio: et a. ad b.g. proportio minor / vt oportet / et ſit d.
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numerus a quo denominatur b. pars aliquota: et
<
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e. a quo denominatur c. pars aliquota: et tūc dico /
<
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tales eſt proportio inter b. et c. qualis inter d.
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et e. </
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<
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xml:space
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">Quod ſic oſtenditur / quia proportio f. que eſt
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a. ad c. excedit proportionem g. que eſt a. ad b. per
<
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proportioneꝫ b. ad c. / vt patet ex primo correlario /
<
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/>
et proportio per quã proportio f. excedit propor-
<
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tionem g. eſt illa que eſt inter denominatiões ſiue
<
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inter termininos a. quibus denominãtur f. et g. pro-
<
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portiones / vt patet ex concluſione: igitur propor-
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tio b. ad c. eſt proportio que eſt inter terminos a
<
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quibus denominatur f. et g. proportiões: et f. et g.
<
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proportiones denominantur a d. et e. numeris a
<
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quibus denominantur b.c. partes aliquote ipſiꝰ
<
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a. / vt patet ex ſecunda petitione igitur: talis eſt ꝓ-
<
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portio inter b. et c. qualis eſt inter d. et e. / quod fuit
<
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probandum. </
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<
s
xml:id
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xml:space
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">Et ſic patet correlariuꝫ.
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xml:id
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xml:space
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">4. correĺ.</
note
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</
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<
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xml:space
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">¶ Sequitur
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quarto / conſtituta naturali ſerie proportionuꝫ
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multipliciū: et conſtituta etiam naturali ſerie pro
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portionum ſuperparticularium: ſecunda ſpecies
<
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proportionis multiplicis excedit primam ſpecieꝫ
<
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per primam ſpeciem proportionis ſuperparticu-
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laris puta per ſexquialterã: et tertia ſpecies mul-
<
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tiplicis excedit ſecundã: per ſecundam ſpeciem ꝓ-
<
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portionis ſuperparticularis: et quarta multipli-
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cis excedit tertiam: per tertiaꝫ ſuperparticularis /
<
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et ſic in infinitum. </
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<
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xml:space
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">Probatur / quia captis primis
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duabus ſpeciebus ꝓportionis multiplicis puta
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dupla et tripla ille denominantur a. numero bina
<
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="
Capitulum quintū.
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rio et ternario / vt conſtat: et tripla excedit duplam
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per proportioneꝫ que eſt inter illos numeros ter-
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narium videlicet et binarium / vt patet in concluſi-
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one: et inter illos eſt prima ſpecies proportionis
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ſuperparticularis / vt patet ex ſecundo capite pri-
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me partis vbi generantur infinite ſpecies propor
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tionis ſuperparticularis ſereatim in naturali ſe
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rie numerorum igitur. </
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<
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">Item captis tripla et qua-
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drupla multiplicibus ille excedunt ſe: per propor
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tionem que eſt .4. ad .3. / vt patet ex concluſiõe: et in-
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ter illos numeros eſt ſecunda ſpecies proportio-
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nis ſuperparticularis / puta ſexquitertia / vt patet
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ex loco preallegato: igit̄̄ correlariū verum quoniã
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eodem modo probabis de aliis.
<
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xml:id
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xml:space
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">5. correĺ.</
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<
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xml:space
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">¶ Sequitur quin
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to / per tot proportiones ſuperparticulares cõ-
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ſequenter / et ſereatim aſſumptas excedit quelibet
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ſpecies multiplicis proportiõis diſtans a. prima
<
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primã ſpeciem multiplicis: per quot vnitates nu-
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merus a quo denominatur illa ſpecies diſtat a
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numero a quo denomīatur prima ſpecies propor
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tionis multiplicis puta dupla. </
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<
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xml:space
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">Et ſic etiam dicen-
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dum eſt de qualibet alia ſpecie mĺtiplici a qua di-
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ſtat per aliquot ſpecies vt proportio quintupla
<
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excedit proportionē duplam per tres ſpecies pro
<
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portionis ſuperparticulares ſereatim ſumptas
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videlicet per proportionem ſexquialteram que eſt
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3. ad .2. et ſexquitertiam que eſt .4. ad .3. et ſexqui-
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quartam que eſt .5. ad .4. </
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<
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facile ex anteriori.
<
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xml:id
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xml:space
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</
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<
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xml:id
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xml:space
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">¶ Sequitur ſexto / vniuerſa-
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lis ſeries proportionum ſuperparticularium in-
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finitam latitudinē proportionis conſtituit. </
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<
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">Pro-
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batur / quia conſtituit infinite magnam proporti-
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onem multiplicem cum proportione dupla: igitur
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talis ſeries in infinitum magna latitudo eſt pro-
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portionis. </
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<
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N13CAA
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xml:space
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">Item talis ſeries proportionum ſuper
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particularium eſt naturalis ſeries numerorum in
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cipiendo a binario: ſed in infinitum magna pro-
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portio eſt alicuius numeri a binarium: igitur infi-
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nitum magna latitudo proportionis eſt natura-
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lis ſeries proportionum ſuperparticularium. </
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<
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">Et
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hoc nota ad capitulum de augmentatione.</
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<
head
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N13CDF
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xml:space
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">Capitulum quintum / in quo reci-
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tatur paucis et impugnatur opinio
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baſani politi de proportione ſiue
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cõmenſurabilitate proportionum.</
head
>
<
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<
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gnanter paripathetici philoſophan-
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tes amputare at reſecare contrari-
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as opinationes: et deinde veras interſerere. </
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<
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">Ideo
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baſani politi opinionem in materia proportio-
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nalitatum ceteris mathematicis aduerſam pre-
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ſenti duximus expugnandam.</
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</
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<
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<
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">Sit igit̄̄ capitalis ſuppoſitio. </
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<
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xml:space
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">Quod
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libet habens ſubduplum eſt duplum ad ſuam me-
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dietatem et ſi ipſum eſt duplum ipſum continet ſu
<
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am medietatem bis adequate. </
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>
<
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N13D08
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xml:space
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">Hec petitio nec
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iuuat eam demonſtrare.</
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</
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<
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<
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xml:space
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">Secunda ſuppoſitio ſiue petitio.
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</
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<
s
xml:id
="
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xml:space
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">Omne duplum ad aliquod continet ipſum vel e-
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quale ei bis tantum: et ſi contineat ipſum pluſquã
<
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bis eſt pluſquam duplum ad illud.</
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>
</
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>
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<
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<
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tur in duplo minus ipſum perdit adequate medi-
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etatem ſui.</
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