Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Secunde partis." file="0044" n="44"/>
                tali ſenſu capitur / vt patet intuenti.</s>
              </p>
              <p xml:id="N14411">
                <s xml:id="N14412" xml:space="preserve">Sꝫ contra / q2 in tali ſenſu capiendo
                  <lb/>
                eã non cõcluditur propoſitum ſed ſolum concludi­
                  <lb/>
                tur /  de qualibet ſpecie proportionis multipli-
                  <lb/>
                cis aliquod indiuiduum eiuſdem ſpeciei non ē cõ-
                  <lb/>
                menſurabile alicui ſuperparticulari, aut ſupraꝑ­
                  <lb/>
                tienti etc. / et adhuc vix id poteſt haberi contra pro­
                  <lb/>
                teruum. </s>
                <s xml:id="N14421" xml:space="preserve">¶ Sed diceret nicholaus /  ſatis ei ē ha-
                  <lb/>
                bere /  vna proportio dupla non eſt commenſura­
                  <lb/>
                bilis alicui proportioni non multiplici rationali /
                  <lb/>
                quoniam cuꝫ omnes duple ſint equales. </s>
                <s xml:id="N1442A" xml:space="preserve">quicquid
                  <lb/>
                non eſt commenſurabile vni certe non eſt commē-
                  <lb/>
                ſurabile alteri. </s>
                <s xml:id="N14431" xml:space="preserve">Et certo credo /  in hoc fundatur
                  <lb/>
                principaliter deductio illarum concluſionū qua-
                  <lb/>
                rum fundamenta ſumuntur ex euclide ſeptimo et
                  <lb/>
                octauo elementorum. </s>
                <s xml:id="N1443A" xml:space="preserve">Notum eni3 eſt /  ſi aliquid
                  <lb/>
                eſt īcommenſurabile vni equalium etiam cuilibet
                  <lb/>
                erit incommenſurabile: quoniam omnia equalia
                  <lb/>
                ex equalibus adequate componuntur.</s>
              </p>
              <p xml:id="N14443">
                <s xml:id="N14444" xml:space="preserve">Sed contra diceret proteruus / quia
                  <lb/>
                dabiles ſunt due proportiones equales et tamen
                  <lb/>
                aliqua proportio eſt pars vnius: et nec illa nec ali­
                  <lb/>
                qua equalis ei eſt pars alterius: igitur non eſt in-
                  <lb/>
                conueniens aliquas duas proportiones eſſe equa­
                  <lb/>
                les: et aliquid eſſe partem vnius et nec illud nec tã­
                  <lb/>
                tum eſſe partem alterius: et per conſequens pari
                  <lb/>
                ratione poſſet dici /  quamuis omnes duple ſint
                  <lb/>
                equales: aliquid tamen eſt pars aliquota vnius /
                  <lb/>
                quod non eſt pars aliquota alterius nec tantum:
                  <lb/>
                quemadmodum aliqua proportio eſt pars alicu-
                  <lb/>
                ius proportionis duple: et tamen nec illa. </s>
                <s xml:id="N1445D" xml:space="preserve">nec ei eq̈­
                  <lb/>
                lia eſt pars alterius duple. </s>
                <s xml:id="N14462" xml:space="preserve">Probatur aſſumptuꝫ
                  <lb/>
                de his duabus duplis quarum vna eſt .8. ad .4. et
                  <lb/>
                altera .2. ad .1. </s>
                <s xml:id="N14469" xml:space="preserve">Nam illa que eſt .8. ad .4. componi-
                  <lb/>
                tur ex ꝓportione ſexquialtera et ſexquitertia que
                  <lb/>
                mediant inter ſua extrema: illa vero que eſt duoꝝ
                  <lb/>
                ad vnum ex nulla ſexquialtera aut ſexquitertia cõ­
                  <lb/>
                ponitur: quoniam nullus numerus mediat inter
                  <lb/>
                extrema illius. </s>
                <s xml:id="N14476" xml:space="preserve">Nec valet dicere /  quamius nõ me­
                  <lb/>
                diat numerus mediat tamen vnitas cum fractio-
                  <lb/>
                ne aliqua: et illud ſufficit: quoniam vnitatis cum
                  <lb/>
                dimidio ad vnitatem eſt proportio ſexquialtera:
                  <lb/>
                </s>
                <s xml:id="N14480" xml:space="preserve">Quoniaꝫ iam tunc haberem /  alicuius ꝓportio-
                  <lb/>
                nis ſexquialtere vnitas eſt alterum extremum / qḋ
                  <lb/>
                ipſe negare videtur. </s>
                <s xml:id="N14487" xml:space="preserve">Et etiam habito illo: iam de-
                  <lb/>
                ſtruitur totus modus procedendi et ꝓbandi illas
                  <lb/>
                concluſiones et etiam quintã. </s>
                <s xml:id="N1448E" xml:space="preserve">Fundatur enim pro­
                  <lb/>
                batio illius quinte concluſionis in hoc:  īter nu­
                  <lb/>
                lius proportionis ſuperparticularis primos nu-
                  <lb/>
                meros reperitur aliqua ꝓportio rationalis que
                  <lb/>
                ſit pars eius. </s>
                <s xml:id="N14499" xml:space="preserve">Modo illud eſt falſum vtendo fra-
                  <lb/>
                ctione vnitatis: inter .5. eī et .6. mediant .5. cū dimi­
                  <lb/>
                dio. </s>
                <s xml:id="N144A0" xml:space="preserve">Item eſto /  inter primos numeros ꝓportio-
                  <lb/>
                nis ſuperparticularis non mediat aliquis nume­
                  <lb/>
                rus mediat tamen inter non primos: et diceret ꝓ-
                  <lb/>
                teruus /  proportio ſuperparticularis inter non
                  <lb/>
                primos numeros componitur ex aliquot rationa­
                  <lb/>
                libus quibus eſt commenſurabilis: et tamen ipſa
                  <lb/>
                proportio inter primos numeros conſtituta non
                  <lb/>
                componitur ex talibus. </s>
                <s xml:id="N144B1" xml:space="preserve">Nec valet dicere /  non eſt
                  <lb/>
                imaginabile /  aliqua duo ſint equalia: et tamen
                  <lb/>
                aliquid ſit pars aliquota vnius et nullum tantuꝫ
                  <lb/>
                ſit pars aliquota alterius. </s>
                <s xml:id="N144BA" xml:space="preserve">quoniam diceret ꝓter­
                  <lb/>
                uus illud non eſſe imaginabile in quantitatibus
                  <lb/>
                continuis: ſed bene eſſe imaginabile in ꝓportioni­
                  <lb/>
                bus quoniam impoſſibile eſt dare duas quantita­
                  <lb/>
                tes cõtinuas equales: et  aliquid ſit pars vnius
                  <lb/>
                ſiue aliquota ſiue non. </s>
                <s xml:id="N144C7" xml:space="preserve">et  nullum tantuꝫ ſit pars
                  <cb chead="Capitulum ſextum"/>
                alterius: et tamen illud datur in proportionibus
                  <lb/>
                </s>
                <s xml:id="N144CE" xml:space="preserve">Duarum enim intelligentiarum ad vnam intelli-
                  <lb/>
                gentiam eſt proportio dupla que non componi-
                  <lb/>
                tur ex ſexquialtera et ſexquitertia nec cum fractio­
                  <lb/>
                ne nec ſine. </s>
                <s xml:id="N144D7" xml:space="preserve">et tamen proportio dupla ei equalis .4.
                  <lb/>
                ad duo componitur ex ſexquialtera et ſexquiter-
                  <lb/>
                tia / vt patet.
                  <note position="right" xlink:href="note-0044-01a" xlink:label="note-0044-01" xml:id="N1450E" xml:space="preserve">Aduerte</note>
                </s>
                <s xml:id="N144E3" xml:space="preserve">¶ Hic tamen tu aduerte /  hee conclu­
                  <lb/>
                ſiones cum demonſtrationibus ſuis dependēt ex
                  <lb/>
                octaua propoſitione octaui elementorum euclidis
                  <lb/>
                que dependet ex .35. ſeptimi, et .14. et .18. et .21. ſepti­
                  <lb/>
                mi et tertia octaui. </s>
                <s xml:id="N144EE" xml:space="preserve">Et ideo difficilis eſt demonſtra­
                  <lb/>
                tio harum concluſionum: quia ex multis depēdēt
                  <lb/>
                  <note position="right" xlink:href="note-0044-02a" xlink:label="note-0044-02" xml:id="N14514" xml:space="preserve">eu. 8. ele.</note>
                </s>
                <s xml:id="N144FA" xml:space="preserve">Dicit tamen euclides in propoſitione allegata 
                  <lb/>
                ſi inter aliquos numeros non primos alicuius ꝓ­
                  <lb/>
                portionis reperiuntur aliqui numeri cõtinuo pro­
                  <lb/>
                portionabiles: totidē inter primos numeros eiuſ­
                  <lb/>
                dem proportionis reperiuntur. </s>
                <s xml:id="N14505" xml:space="preserve">Et ideo tu ipſe ef-
                  <lb/>
                ficatiores demonſtrationes inquire.</s>
              </p>
              <p xml:id="N1451A">
                <s xml:id="N1451B" xml:space="preserve">Octaua concluſio. </s>
                <s xml:id="N1451E" xml:space="preserve">Si fuerint tres
                  <lb/>
                termini continuo proportionabiles geometri-
                  <lb/>
                ce erit proportio extremi ad extremum dupla ad
                  <lb/>
                vtrã intermediam. </s>
                <s xml:id="N14527" xml:space="preserve">et ſi fuerint .4. tripla, ſi .5. q̈-
                  <lb/>
                drupla: et ſic in infinitum. </s>
                <s xml:id="N1452C" xml:space="preserve">ſemper vno minus. </s>
                <s xml:id="N1452F" xml:space="preserve">hoc
                  <lb/>
                eſt ſi fuerint decem termini non erit ꝓportio decu­
                  <lb/>
                pla extremi ad extremum: ſed noncupla. </s>
                <s xml:id="N14536" xml:space="preserve">Proba-
                  <lb/>
                tur: quoniam ſi ſunt tres termini continuo ꝓpor-
                  <lb/>
                tionabiles: reperientur ibi due ꝓportiones equa­
                  <lb/>
                les ex quibus adequate componitur ꝓportio ex-
                  <lb/>
                tremi ad extremum: et ſi quatuor tres. </s>
                <s xml:id="N14541" xml:space="preserve">et ſi quin
                  <lb/>
                quatuor / et ſic conſequenter. </s>
                <s xml:id="N14546" xml:space="preserve">Modo omne compo-
                  <lb/>
                ſitum ex duobus equalibus adequate eſt duplum
                  <lb/>
                ad quodlibet illorum, et ex tribus tripluꝫ, et ſic cõ­
                  <lb/>
                ſequenter / vt patet ex quinta ſuppoſitione quarti
                  <lb/>
                capitis huius partis: igitur cõcluſio vera:
                  <note position="right" xlink:href="note-0044-03a" xlink:label="note-0044-03" xml:id="N14569">
                    <s xml:id="N1456D" xml:space="preserve">eu. 5. ele.
                      <lb/>
                    </s>
                    <s xml:id="N14571" xml:space="preserve">ior. 2. ele.
                      <lb/>
                    </s>
                    <s xml:id="N14575" xml:space="preserve">Ne hoc
                      <lb/>
                    p̄tereas.</s>
                  </note>
                </s>
                <s xml:id="N14556" xml:space="preserve">¶ Et hec
                  <lb/>
                eſt decima diffinitio quinti elementorum euclidis
                  <lb/>
                et quinta diffinitio ſecundi elementorum iordani
                  <lb/>
                </s>
                <s xml:id="N1455E" xml:space="preserve">¶ Et aduerte /  quotienſcun allego euclidē: ſem­
                  <lb/>
                per vtor noua traductione. Bartholomei3 am-
                  <lb/>
                berti.</s>
              </p>
              <p xml:id="N1457A">
                <s xml:id="N1457B" xml:space="preserve">Nona concluſio </s>
                <s xml:id="N1457E" xml:space="preserve">Nulla proportio ra­
                  <lb/>
                tionalis habet ſubduplam rationalem. </s>
                <s xml:id="N14583" xml:space="preserve">niſi habe­
                  <lb/>
                at numerū mediū ꝓportionabilem inter ſua extre­
                  <lb/>
                ma: et ſi non habet talem numerum non habet ſub­
                  <lb/>
                quadruplam proportionem rationalem, nec ſub­
                  <lb/>
                octuplam: nec ſubſexdecuplam: et ſic in infinitum
                  <lb/>
                procedendo per numeros pariter. </s>
                <s xml:id="N14590" xml:space="preserve">Proba­
                  <lb/>
                tur prima pars huius concluſionis: quia ſi nõ de-
                  <lb/>
                tur oppoſitum videlicet /  aliqua proportio ha-
                  <lb/>
                beat ſubduplam rationaleꝫ que non habet nume­
                  <lb/>
                rum medium ꝓportionabilem inter ſua extrema:
                  <lb/>
                et ſit illa a. / et arguo ſic / a. proportio habet ꝓpor-
                  <lb/>
                tionem ſubduplam rationalem que ſit f. gratia ex­
                  <lb/>
                empli: igitur a. proportio componitur ex duplici
                  <lb/>
                f: adequate et per conſequēs vna illaruꝫ f. erit ma­
                  <lb/>
                ioris extremi ipſius a. ad aliquem numerum inter­
                  <lb/>
                medium: et altera eiuſdem numeri intermedii ad
                  <lb/>
                aliud extremum minus eiuſdem a. ꝓportionis: et
                  <lb/>
                per conſequens ille numerus intermedius erit me­
                  <lb/>
                dio loco proportionabilis / vt patet ex diffinitiõe
                  <lb/>
                numeri medio loco proportionabilis / quod eſt op­
                  <lb/>
                poſitum dati. </s>
                <s xml:id="N145B1" xml:space="preserve">Iam probatur ſecunda pars: quo-
                  <lb/>
                niam ſi inter terminos date ꝓportionis rationa­
                  <lb/>
                lis non fuerit numerus qui ſit medium proportio­
                  <lb/>
                nale: iam ibi non reperiuntur quin numeri cõti-
                  <lb/>
                nuo proportionabiles geometrice: et ſi non ſunt
                  <lb/>
                ibi quin numeri cõtinuo proportionabiles geo­
                  <lb/>
                metrice: iam extremi ad extremum non erit ꝓpor-
                  <lb/>
                tio quadrupla ad aliquam proportionem ratio- </s>
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