Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Secunde partis." file="0046" n="46"/>
                quia aggregatum ex aliquo et medietate eiꝰ ē ſex­
                  <lb/>
                quialterum ad illud / vt conſtat ex diffinitione ſex-
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                quialteri. </s>
                <s xml:id="N14764" xml:space="preserve">Et iſto modo inuenitur octuplam eē ſex­
                  <lb/>
                quialteram ad quadruplam. </s>
                <s xml:id="N14769" xml:space="preserve">Si vero inueſtigare
                  <lb/>
                et ſcire velis an q̈drupla habeat ſexquiquartam
                  <lb/>
                ſcias primo ꝑ doctrinam ſecundi correlarii: an ip­
                  <lb/>
                ſa proportio quadrupla habeat ſubquadruplaꝫ
                  <lb/>
                rationalem: et ſi ſic concludas /  habet ſexquiq̈r-
                  <lb/>
                tam rationalem: quoniam reperta quarta ipſius
                  <lb/>
                quadruple ad dandam ſexquiquartam ad ipſam
                  <lb/>
                quadruplam nihil aliud oportet quaꝫ addere ipſi
                  <lb/>
                quadruple ſuam quartam: et tunc aggregatuꝫ ex
                  <lb/>
                ipſa quadrupla et ſua quarta rationali ſe habet
                  <lb/>
                ad ipſaꝫ quadrumplam in proportiõe ſexquiquar­
                  <lb/>
                ta. </s>
                <s xml:id="N14782" xml:space="preserve">Continet enim illud aggregatum ipſam qua-
                  <lb/>
                druplam et vnam quartam eius adequate. </s>
                <s xml:id="N14787" xml:space="preserve">Et iſto
                  <lb/>
                modo inuenitur trigecuplam ſecūdam eſſe ſexqui­
                  <lb/>
                quartam ad ſexdecuplam. </s>
                <s xml:id="N1478E" xml:space="preserve">Et iſto modo in quali-
                  <lb/>
                bet proportione rationali īueſtigare poteris: an
                  <lb/>
                habeat ſexquioctauam, ſexquiſexdecimam, et ſic
                  <lb/>
                conſequēter rationales. </s>
                <s xml:id="N14797" xml:space="preserve">Et ſic patet correlarium
                  <lb/>
                  <note position="left" xlink:href="note-0046-01a" xlink:label="note-0046-01" xml:id="N14839" xml:space="preserve">4. correl.</note>
                </s>
                <s xml:id="N147A1" xml:space="preserve">¶ Ex quo ſequitur quarto /  ſi aliqua ꝓportio ra­
                  <lb/>
                tionalis non habet ſubduplam rationalem: ipſa
                  <lb/>
                non habet ſexquialteram rationalem, nec ſexqui­
                  <lb/>
                q̈rtã: nec ſexquioctauam: nec ſexquiſexdecimam: et
                  <lb/>
                ſic conſequenter. </s>
                <s xml:id="N147AC" xml:space="preserve">Probatur / quia ſi talis ꝓportio
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                non habeat ſubduplam rationaleꝫ: ſequitur /  nõ
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                habet numerum qui ſit medium ꝓportionale īter
                  <lb/>
                ſua extrema: et ſi nõ hꝫ numerū mediū etc. / ſequit̄̄ 
                  <lb/>
                 non habet ſubquadruplam, nec ſuboctuplam,
                  <lb/>
                nec ſubſexdecuplam rationalem / et ſic in infinituꝫ
                  <lb/>
                aſcendendo per numeros pariter pares / vt patet
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                ex nona concluſione huius: et ſi non habet ſubdu-
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                plam, nec ſubquadruplam: nec ſuboctuplam ra-
                  <lb/>
                tionales: et ſic conſequenter: iam manifeſtum eſt /
                  <lb/>
                 non habet ſexquialteram rationalem: nec ſex-
                  <lb/>
                quiquartam: nec ſexquioctauam: et ſic ſine fine / vt
                  <lb/>
                patet ex probatione precedentis correlarii. </s>
                <s xml:id="N147C7" xml:space="preserve">Et ſic
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                ſi data proportio rationalis nõ habet ſubduplaꝫ
                  <lb/>
                rationalem: ipſa non habet ſexquialteram ratio­
                  <lb/>
                nalem: nec ſexquiquartaꝫ: nec ſexquioctauã etc. / qḋ
                  <lb/>
                fuit probandum. </s>
                <s xml:id="N147D2" xml:space="preserve">Et ſic patet correlarium.
                  <note position="left" xlink:href="note-0046-02a" xlink:label="note-0046-02" xml:id="N1483F" xml:space="preserve">5. correl.</note>
                </s>
                <s xml:id="N147DA" xml:space="preserve">¶ Se-
                  <lb/>
                quitur quinto /  ſi aliqua proportio ꝓpoſita non
                  <lb/>
                habuerit ſubduplam rationalem: ipſa non habe­
                  <lb/>
                bit duplam ſexquialteram rationalem nec duplã
                  <lb/>
                ſexquiquartam nec ſuprapartienteꝫ quartas, nec
                  <lb/>
                aliquam ſuprapartientem denominatam ab vni­
                  <lb/>
                tate et partibus aliquotis denominatis a nume-
                  <lb/>
                ro pariter pari: nec aliquam multiplicē ſuperpar­
                  <lb/>
                ticularem, aut multiplicē ſuprapartientem deno­
                  <lb/>
                minatã a numero et a parte vel partibus aliquo-
                  <lb/>
                tis que denominantur a numeris pariter paribꝰ
                  <lb/>
                </s>
                <s xml:id="N147F2" xml:space="preserve">Patet hoc correlarium facile: quia ſi data ꝓpor­
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                tio non habuerit ſubduplam rationalem: iam nõ
                  <lb/>
                habet illas partes aliquotas rationales deno-
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                minatas a numeris pariter paribus: vt patet ex
                  <lb/>
                quarto correlario: et ſi non habet illas partes ali­
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                quotas que ſunt ꝓportiones rationales: iam non
                  <lb/>
                habet illas proportiones rationales denomina-
                  <lb/>
                tas ab illis partibus / vt conſtat.
                  <note position="left" xlink:href="note-0046-03a" xlink:label="note-0046-03" xml:id="N14845" xml:space="preserve">6. correl.</note>
                </s>
                <s xml:id="N14808" xml:space="preserve">¶ Ex quo ſequi-
                  <lb/>
                tur ſexto /  nec tripla, nec dupla, habent ꝓportio­
                  <lb/>
                nē ſexquialterã: ſexquiquartam: ſexquioctauam:
                  <lb/>
                duplã ſupratripartientē quartas rationalem: et
                  <lb/>
                ſic de multis aliis. </s>
                <s xml:id="N14813" xml:space="preserve">Patet / quia neutra illarum ha­
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                bet ſubduplam rationalem: vt patet ex primo cor­
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                relario: igitur neutra illarum habet ſexquialterã
                  <lb/>
                ſexquiquartam etc. / vt patet ex īmediate preceden-
                  <lb/>
                ti. </s>
                <s xml:id="N1481E" xml:space="preserve">Inferas tu ſimilia correlaria particularia ex
                  <lb/>
                dictis.</s>
              </p>
              <cb chead="Capitulum ſextum"/>
              <p xml:id="N1484D">
                <s xml:id="N1484E" xml:space="preserve">Undecima concluſio. </s>
                <s xml:id="N14851" xml:space="preserve">Nulla propor-
                  <lb/>
                tio rõnalis ſe habet ī aliqua proportiõe multipli­
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                ci ad aliquam rationalem niſi inter primos nūe-
                  <lb/>
                ros eius reperiantur tot numeri cõtinuo ꝓportio­
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                nabiles computatis etiam extremis vno plꝰ ade-
                  <lb/>
                quate: quotus eſt numerus a quo denomīatur da­
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                ta ꝓportio multiplex. </s>
                <s xml:id="N14860" xml:space="preserve">Exemplum. </s>
                <s xml:id="N14863" xml:space="preserve">vt ſi velis inue-
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                ſtigare et ſcire vtrum ꝓportio quadrupla ſe habe­
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                at in ꝓportione dupla ad aliquam ꝓportioneꝫ
                  <lb/>
                rationalem: conſidera primum a quo numero de­
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                nominatur proportio dupla: et īuenies /  a bina­
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                rio iuxta doctrinam primi correlarii ſecunde ſup­
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                poſitionis quarti capitis huius: tunc capias pri­
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                mos numeros eius qui ſunt .4. et .1: et vide ſi inue-
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                nias ibi tres numeros continuo ꝓportionabiles
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                eadem ꝓportione cõputatis extremis: et ſi ſic dico /
                  <lb/>
                 ꝓportio quadrupla ſe habet in ꝓportione du-
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                pla ad aliquaꝫ rationalem. </s>
                <s xml:id="N1487C" xml:space="preserve">Si enim ibi ſunt tres
                  <lb/>
                numeri continuo ꝓportionabiles computatis ex­
                  <lb/>
                tremis: iam illa ꝓportio quadrupla que eſt extre-
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                mi ad extremum eſt dupla ad vtrã interdiarum:
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                vt patet ex octaua concluſione: et ſi velis ſcire an
                  <lb/>
                quadrupla ſit tripla ad aliquam ꝓportionem ra­
                  <lb/>
                tionalem: quia tripla denominatur a numero ter­
                  <lb/>
                nario. </s>
                <s xml:id="N1488D" xml:space="preserve">videas vtrum inter primos numeros ꝓpor­
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                tionis quadruple reperiantur tres nūeri vno plꝰ
                  <lb/>
                puta quatuor continuo ꝓportionabiles aliqua ꝓ­
                  <lb/>
                portione: et ſi ſic: tunc quadrupla ſe habet in pro-
                  <lb/>
                portione tripla ad aliquam ꝓportionē rationalē
                  <lb/>
                puta ad quãlibet illarum conſtitutarum inter ali­
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                quos ex illis numeris continuo ꝓportionabilibꝰ
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                et īmediatis: et quia tu non inuenies inter primos
                  <lb/>
                numeros ꝓportionis quadruple quatuor nume-
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                ros continuo ꝓportionabiles computatis extre-
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                mis: concludas /  quadrupla nõ habet ſubtriplã
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                rationalem. </s>
                <s xml:id="N148A6" xml:space="preserve">Probatur hec concluſio. </s>
                <s xml:id="N148A9" xml:space="preserve">q2 ſi data ꝓ­
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                portio rationalis que ſit a. ſe habeat in aliqua ꝓ­
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                portione multiplici ad aliquam proportioneꝫ ra­
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                tionalem que ſit b. / ſequitur /  a. aliquoties conti-
                  <lb/>
                net b. adequate / et ſic b. erit pars aliquota ipſius
                  <lb/>
                a denominata a numero a quo denominatur pro­
                  <lb/>
                portio multiplex in qua a. ſe habet ad b. / vt puta ſi
                  <lb/>
                a. ſe habet ad b: in proportione quadrupla erit b.
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                vna quarta ipſius a. et ſic erit b. pars aliquota de­
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                nominata a numero quaternario a quo denomi-
                  <lb/>
                natur ꝓportio illa multiplex puta quadrupla in
                  <lb/>
                qua a. ſe habet ad b: et ſi ſic iam neceſſe eſt  b. re-
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                periatur inter aliquos numeros ipſius a. toties
                  <lb/>
                quoties eſt numerus a quo denominatur talis ꝓ-
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                portio multiplex in qua a. ſe habet ad b. et ſi ſic iã
                  <lb/>
                inter terminos ipſius a. computatis extremis re-
                  <lb/>
                perientur tot nūeri quotus eſt ille numerus a quo
                  <lb/>
                denominatur data ꝓportio multiplex in qua a. ſe
                  <lb/>
                habet ad b. vno plus: quoniam ſemper termini ſi­
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                ue numeri continuo ꝓportionabiles ſunt vno plu­
                  <lb/>
                res proportionibus inter ipſos ad inuentis / vt ptꝫ
                  <lb/>
                ex octaua concluſione huius: et ex conſequēti ſi nõ
                  <lb/>
                fuerint reperti tot numeri continuo ꝓportionabi-
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                les inter aliquos numeros ipſius proportionis a.
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                quotus eſt numerus a quo denominatur propor-
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                tio multiplex in qua ponitur a. ſe habere ad b. / di-
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                co /  tūc b. non eſt ꝓportio rationalis nec a. ſe ha­
                  <lb/>
                bet in tali ꝓportione multiplici ad aliquam pro-
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                portionem rationalem. </s>
                <s xml:id="N148E4" xml:space="preserve">Probatur hec conſequē-
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                tia / quia ſi ſe haberet ad b. proportioneꝫ rationa­
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                lem in tali ꝓportione multiplici: iam aliquoties
                  <lb/>
                componeretur ex ipſa b. ꝓportione rationali et ꝑ
                  <lb/>
                conſequens aliquoties reperiretur b. inter nume-
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                ros eius: puta toties quotus ē numerus a quo de- </s>
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