Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

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                  <pb chead="Secūde partis" file="0047" n="47"/>
                nominatnr data ꝓportio multiplex: et ſi ſic iã in-
                  <lb/>
                ter terminos eius computatis extremis reperiren­
                  <lb/>
                tur tot numeri continuo ꝓportionabiles quotus
                  <lb/>
                eſt numerus a quo denominatur dicta proportio
                  <lb/>
                multiplex: puta quoties a. cõtinet b. vno plus. </s>
                <s xml:id="N148FE" xml:space="preserve">igi­
                  <lb/>
                tur ex oppoſito: ſi non reperiantur tot numeri cõ-
                  <lb/>
                putatis extremis iam a. non ſe habet in tali ꝓpor­
                  <lb/>
                tione multiplici ad b. ꝓportionem rationalem.</s>
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              <note position="left" xml:id="N14907" xml:space="preserve">nota.</note>
              <p xml:id="N1490B">
                <s xml:id="N1490C" xml:space="preserve">¶ Utrum autē inter aliquos numeros date ꝓpor­
                  <lb/>
                tionis a. reperiantur tot numeri continuo ꝓpor-
                  <lb/>
                tionabiles computatis extremis vno plus quotꝰ
                  <lb/>
                eſt numerus a quo denominatur proportio multi­
                  <lb/>
                plex in qua ponitur a. ſe habere ad b. videndū eſt
                  <lb/>
                vtrum inter primos numeros eius inueniant̄̄ tot
                  <lb/>
                numeri continuo proportionabiles: et ſi ſic conclu­
                  <lb/>
                das /  inter numeros ipſius a. reperiuntur tot nu­
                  <lb/>
                meri continuo ꝓportionabiles: et ſi non inuenian­
                  <lb/>
                tur tot inter primos numeros date ꝓportionis:
                  <lb/>
                dicas /  inter nullos numeros eius reperiunt̄̄ tot
                  <lb/>
                numeri continuo ꝓportinoabiles computatis ex­
                  <lb/>
                tremis. </s>
                <s xml:id="N14927" xml:space="preserve">Patet hec conſequentia / et deductio tota
                  <lb/>
                ex octaua ꝓpoſitione octaui elementorum eucli-
                  <lb/>
                dis in qua habetur /  ſi inter duos numeros ceci-
                  <lb/>
                derint aliqui numeri continuo ꝓportionabiles:
                  <lb/>
                inter quoſcun duos in eadem ꝓportione ſe ha-
                  <lb/>
                bentes cadent tot numeri continuo ꝓportionabi­
                  <lb/>
                les eadem ꝓportione qua ꝓportionautur alii. </s>
                <s xml:id="N14936" xml:space="preserve">ex
                  <lb/>
                qua immediate infertur /  ſi inter duos numeros
                  <lb/>
                ſe habentes in ꝓportio a. ceciderint aliqui nume-
                  <lb/>
                ri continuo ꝓportionabiles ꝓportiõe que eſt vna
                  <lb/>
                tertia: aut vna quarta: aut vna quinta: ipſius a. in­
                  <lb/>
                ter primos numeros ipſius a. tot numeri cadēt ꝓ­
                  <lb/>
                portionabiles eadeꝫ ꝓportione que ſit tertia aut
                  <lb/>
                quarta: aut quinta ipſius a. / igitur ex oppoſito cõ­
                  <lb/>
                ſequentis ſi inter primos numeros a. proportio-
                  <lb/>
                nis non reperiantur aliqui numeri continuo pro­
                  <lb/>
                portionabiles ꝓportione que eſt vna tertia: vna
                  <lb/>
                quarta: quinta: ipſius a. et c. nec inter aliquos nūe­
                  <lb/>
                ros ipſius a. reperientur: quod fuit oſtendendum:
                  <lb/>
                </s>
                <s xml:id="N14952" xml:space="preserve">Et ſic patet concluſio.
                  <note position="left" xlink:href="note-0047-01a" xlink:label="note-0047-01" xml:id="N14B1B" xml:space="preserve">1. correl.</note>
                </s>
                <s xml:id="N1495A" xml:space="preserve">¶ Ex quo ſequitur primo. / 
                  <lb/>
                ꝓportio dupla ad nullam ꝓportionem rationa-
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                lem ſe habet in ꝓportione dupla: aut tripla. aut
                  <lb/>
                quadrupla: aut in aliqua alia multiplici: nec quin­
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                tupla, nec ſextupla etc. </s>
                <s xml:id="N14965" xml:space="preserve">Probatur / quia inter pri-
                  <lb/>
                mos numeros ꝓportionis duple nullus numerus
                  <lb/>
                reperitur (computamus enim vnitatem pro nume­
                  <lb/>
                ro). </s>
                <s xml:id="N1496E" xml:space="preserve">Item inter primos numeros proportionis
                  <lb/>
                quintuple qui ſunt .5. et .1. non reperiuntur aliqui
                  <lb/>
                numeri continuo ꝓportionabiles adequate com­
                  <lb/>
                putatis extremis / vt conſtat. </s>
                <s xml:id="N14977" xml:space="preserve">Et ſic patet etiam de
                  <lb/>
                ſextupla. </s>
                <s xml:id="N1497C" xml:space="preserve">Patet igitur correlarium.
                  <note position="left" xlink:href="note-0047-02a" xlink:label="note-0047-02" xml:id="N14B21" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N14984" xml:space="preserve">¶ Sequitur
                  <lb/>
                ſecundo /  nulla ꝓportio ſuperparticularis ſe ha­
                  <lb/>
                bet in aliqua ꝓportione multiplici ad aliquam ꝓ­
                  <lb/>
                portionem rationalem. </s>
                <s xml:id="N1498D" xml:space="preserve">Patet / quia inter cuiuſli­
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                bet ſuperparticularis primos terminos nullꝰ re-
                  <lb/>
                peritur numerus: igitur.
                  <note position="left" xlink:href="note-0047-03a" xlink:label="note-0047-03" xml:id="N14B27" xml:space="preserve">3. correl.</note>
                </s>
                <s xml:id="N14999" xml:space="preserve">¶ Sequitur tertio /  pro­
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                poſita quauis proportione rationali inueſtigare
                  <lb/>
                poſſumus an habeat aliquam ꝓportionem ratio­
                  <lb/>
                nalem que ſe habeat ad ipſam in ꝓportione ſexq̇-
                  <lb/>
                altera: ſexquitertia: ſexquiquarta etc. / vt ꝓpoſita
                  <lb/>
                ꝓportione dupla: videre an ſit aliqua ꝓportio ra­
                  <lb/>
                tionalis que ſe habeat ad ipſam duplam in pro-
                  <lb/>
                portione ſexquialtera, ſexquitertia, aut in aliqua
                  <lb/>
                alia ſuperparticulari. </s>
                <s xml:id="N149AC" xml:space="preserve">Ad quod inueſtiganduꝫ et
                  <lb/>
                ſciendum videndum eſt an inter primos numeros
                  <lb/>
                ꝓportiouis duple aut cuiuſuis alterius rationa-
                  <lb/>
                lis ſint tres numeri continuo ꝓportionabiles cõ-
                  <lb/>
                putatis extremis: et ſi ſic: talis ꝓportio habet me­
                  <lb/>
                dietatem rationalem: et per conſequens ſexquial­
                  <cb chead="Capitulum ſextum"/>
                teram rationalem ad ipſam. </s>
                <s xml:id="N149BC" xml:space="preserve">Addendo enī et me-
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                dietatem ſui conſtituetur ſexquialtera rationalis
                  <lb/>
                ad ipſaꝫ. </s>
                <s xml:id="N149C3" xml:space="preserve">Et ſi inter primos numeros eius compu­
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                tatis extremis inueniantur quatuor numeri conti­
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                nuo ꝓportionabiles: ipſa habebit tertiam ratio­
                  <lb/>
                nalem et per conſequens ſexquitertiam rationa-
                  <lb/>
                lem ad ſeipſam: et ſi reperiuntur .5. numeri conti-
                  <lb/>
                nuo ꝓportionabiles computatis extremis ip̄a ha­
                  <lb/>
                bebit quartam rationalem: et per conſequens ſex­
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                quiquartam rationalem / et ſic conſequenter. </s>
                <s xml:id="N149D4" xml:space="preserve">Et
                  <lb/>
                ſic patet correlarium.
                  <note position="right" xlink:href="note-0047-04a" xlink:label="note-0047-04" xml:id="N14B2D" xml:space="preserve">4. correl.</note>
                </s>
                <s xml:id="N149DE" xml:space="preserve">¶ Sequitur quarto /  ꝓpo­
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                ſita quauis ꝓportione rationali: inquirere et ſci-
                  <lb/>
                re poterimus an habeat aliquam ſuprapartien-
                  <lb/>
                tem, multiplicem ſuperparticulareꝫ, vel multipli­
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                cem ſuprapartientem, rationales. </s>
                <s xml:id="N149E9" xml:space="preserve">vt ꝓpoſita pro­
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                portione octupla īueſtigare poterimus et ſcire ex
                  <lb/>
                dictis an habeat ſuprabipartientem tertias ſu-
                  <lb/>
                prapartientem quartas rationales etc. </s>
                <s xml:id="N149F2" xml:space="preserve">Ad quod
                  <lb/>
                ſciendum et inueſtigandum: conſiderandum ē an
                  <lb/>
                data proportio rationalis habeat illam partem
                  <lb/>
                aliquotam rationalem: hoc eſt an aliqua propor­
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                tio rationalis ſit tota pars aliquota eius quota
                  <lb/>
                eſt illa a qua denominatur dicta proportio ſupra­
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                partiens, ant multiplex ſuperparticularis, aut
                  <lb/>
                multiplex ſuprapartiens: quod inueſtigari et ſciri
                  <lb/>
                debet ex vndecima concluſione: et ſi repperias / 
                  <lb/>
                habet proportionem aliquam rationalem que ſit
                  <lb/>
                talis pars aliquota eius: tunc manifeſtum ē /  ha­
                  <lb/>
                bet proportionem rationalem que denominatur
                  <lb/>
                a tali parte aliquota vel talibus partibus aliquo­
                  <lb/>
                tis (quod dico ꝓpter ſuprapartientes) ſi vero nõ:
                  <lb/>
                tunc manifeſtum eſt illam proportionem rationa­
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                lem propoſitam non habere proportionem ratio­
                  <lb/>
                nalem denominatam a tali parte aliquota vel ta­
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                libus partibus. </s>
                <s xml:id="N14A17" xml:space="preserve">Probatur hoc demonſtratione
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                particulari que equiualebit vniuerſali. </s>
                <s xml:id="N14A1C" xml:space="preserve">Data em̄
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                ꝓportione ſexdecupla volo inueſtigare et ſcire an
                  <lb/>
                habeat proportionem ſupratripartientem quar-
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                tas ad quod inueſtigandum conſiderabo ex doc-
                  <lb/>
                trina vndecime concluſionis an talis ꝓportio ſex­
                  <lb/>
                decupla habeat ſubquadruplam rationaleꝫ que
                  <lb/>
                ſit vna quarta eius: et inuento  ſic eo /  inter ter­
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                minos eius computatis extremis inueniuntur
                  <lb/>
                quin numeri continuo ꝓportionabiles ꝓportio­
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                ne dupla: aſſeuerabo conſtanter illam proportio­
                  <lb/>
                nem habere proportionem rationalem ſupertri-
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                partientem quartas: et multiplicem ſexquiquar-
                  <lb/>
                tam et multiplicem ſupratripartientem quartas
                  <lb/>
                rationales. </s>
                <s xml:id="N14A39" xml:space="preserve">Quod ſic monſtratur </s>
                <s xml:id="N14A3C" xml:space="preserve">Nam ſi ſupra il­
                  <lb/>
                lam proportionem ſexdecuplam que eſt .16. ad .1.
                  <lb/>
                addantur tres proportiones duple: tunc aggre-
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                gatum ex ſexdecupla et illis tribus duplis ſuꝑ ad­
                  <lb/>
                ditis qualis eſt proportio .128. ad .1. ſe habebit ad
                  <lb/>
                proportionem ſexdecuplam in proportiõe ſupra-
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                tripartiente quartas. </s>
                <s xml:id="N14A4B" xml:space="preserve">Continet enim ſexdecu-
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                plam et tres quartas eius. </s>
                <s xml:id="N14A50" xml:space="preserve">Item triplando illam
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                proportionem ſexdecuplam / et addendo vnam ſui
                  <lb/>
                quartam habebis ꝓportionem triplam ſexquiq̈r­
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                tam ad ſexdecuplam: et addendo ei duas quartas
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                habebis triplam ſexquialteram: et addendo ſuꝑ
                  <lb/>
                illam triplatam .3. quartas habebis triplam ſu-
                  <lb/>
                pratripartientem quartas rationalem ad ſexde-
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                cuplam. </s>
                <s xml:id="N14A61" xml:space="preserve">Omnia iſta patet ex diffinitionibus ſu-
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                prapartiētis, multiplicis ſuperparticularis. </s>
                <s xml:id="N14A66" xml:space="preserve">aut
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                multiplicis ſuprapartientis. </s>
                <s xml:id="N14A6B" xml:space="preserve">hoc addito /  cuili-
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                bet proportioni rationali addi poteſt queuis alia
                  <lb/>
                rationalis: aggregato ex ipſis manente rationa­
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                li proportione. </s>
                <s xml:id="N14A74" xml:space="preserve">Ex quibuſcnn enim rationalibꝰ
                  <lb/>
                et quotcun: rationalis componitur: q2 alias in </s>
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