Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N14B3F" level="3" n="7" type="chapter" type-free="capitulum">
              <p xml:id="N14C01">
                <s xml:id="N14C35" xml:space="preserve">
                  <pb chead="Secunde partis" file="0049" n="49"/>
                tur aliquis numeris cū fractione vel ſine habens
                  <lb/>
                ſe in eadem proportione ad illud maius extremū:
                  <lb/>
                vt patet ex tertia ſuppoſitione: et tūc illius nume-
                  <lb/>
                ri ad minimū numerū erit ꝓportio dupla ad illaꝫ
                  <lb/>
                ſuperparticularē: q2 ibi erūt tres termini cõtinuo
                  <lb/>
                ꝓportionabiles .etc̈. </s>
                <s xml:id="N14C49" xml:space="preserve">Et iſto modo poteris cõſttue-
                  <lb/>
                re .5. terminos .6.7. continuo ꝓportionabiles: illa
                  <lb/>
                ꝓportione ſuperparticulari data: et ſic in infinitū /
                  <lb/>
                igit̄̄ dabitur ad eam quadrupla, quītupla, ſextu-
                  <lb/>
                pla rationalis: et ſic in infinitū. </s>
                <s xml:id="N14C54" xml:space="preserve">Et eodē modo pro­
                  <lb/>
                babis de quocū genere ꝓportionū rationaliuꝫ
                  <lb/>
                </s>
                <s xml:id="N14C5A" xml:space="preserve">Et ſic patet concluſio.</s>
              </p>
              <p xml:id="N14C5D">
                <s xml:id="N14C5E" xml:space="preserve">Secūda cõcluſio. </s>
                <s xml:id="N14C61" xml:space="preserve">Quãuis quelibet
                  <lb/>
                ꝓportio rationalis in qualibet ꝓportione multi-
                  <lb/>
                plici ab aliqua ꝓportione ratiõali excedatur: ita­
                  <lb/>
                 quelibet ꝓportio rationalis habeat duplã, tri-
                  <lb/>
                plam, quadruplã, rationales / et ſic in infinitū: ni-
                  <lb/>
                chilominus nõ quelibet ꝓportio ratiõalis habet
                  <lb/>
                ſubduplã, ſubtriplã, ſubquadruplã, rationales.
                  <lb/>
                etc̈. </s>
                <s xml:id="N14C72" xml:space="preserve">Prima pars huiꝰ concluſionis patet ex priori
                  <lb/>
                concluſione: et ſecunda ꝓbatur: quia ꝓportio du-
                  <lb/>
                pla non habet ſubduplã rationalē, nec ſubtriplã,
                  <lb/>
                nec ſubquadruplã .etc̈. / vt patet ex doctrina vnde-
                  <lb/>
                cime concluſionis precedentis capitis: igitur non
                  <lb/>
                quelibet ꝓportio rationalis habet ſubduplã ſub­
                  <lb/>
                triplã, ſubq̈druplã ratiõales .etc̈. </s>
                <s xml:id="N14C81" xml:space="preserve">Ptꝫ igit̄̄ ↄ̨cluſio</s>
              </p>
              <p xml:id="N14C84">
                <s xml:id="N14C85" xml:space="preserve">Tertia cõcluſio. </s>
                <s xml:id="N14C88" xml:space="preserve">Aliqua ꝓportio ra-
                  <lb/>
                tionalis eſt dupla, tripla, quadrupla, et ſic in infi­
                  <lb/>
                nitū alicui ꝓportioni irratiõali. </s>
                <s xml:id="N14C8F" xml:space="preserve">Probatur / quia
                  <lb/>
                ꝓportio dupla eſt huiuſmodi / igitur. </s>
                <s xml:id="N14C94" xml:space="preserve">Antecedens
                  <lb/>
                ꝓbatur / quia ꝓportio dupla habet medietatē ter­
                  <lb/>
                tiam, quartã, quintã .etc̈. / vt patet ex quinta ſuppo­
                  <lb/>
                ſitione: et ad medietatē ſui eſt dupla, et ad tertiaꝫ
                  <lb/>
                tripla, et ſic in infinitū / vt patet ex quarta ſuppo-
                  <lb/>
                ſitione: et nec eius medietas, nec eius tertia, et ſic
                  <lb/>
                in infinitū ſunt ꝓportiones rationales / vt patet ex
                  <lb/>
                ꝓbatione precedentis cõcluſionis: igit̄̄ ſunt ꝓpor­
                  <lb/>
                tiões irratiõales: igit̄̄ ipſa ꝓportio dupla eſt du-
                  <lb/>
                pla, tripla, quadrupla, et ſic in infinitū alicui pro­
                  <lb/>
                portioni irrationali / quod fuit probandum.</s>
              </p>
              <p xml:id="N14CAB">
                <s xml:id="N14CAC" xml:space="preserve">Quarta cõcluſio. </s>
                <s xml:id="N14CAF" xml:space="preserve">Quelibet ꝓportio
                  <lb/>
                rationalis eſt cõmenſurabilis alicui proportioni
                  <lb/>
                irrationali. </s>
                <s xml:id="N14CB6" xml:space="preserve">Probatur hec concluſio / qm̄ nulla ꝓ-
                  <lb/>
                portio ratiõalis habet quãlibet ſui partē aliquo-
                  <lb/>
                tam rationalē ꝓportionē: igitur quelibet eſt com­
                  <lb/>
                menſurabilis alicui rationali. </s>
                <s xml:id="N14CBF" xml:space="preserve">Patet cõſequētia
                  <lb/>
                ſuppoſita cõſtantia: qm̄ quelibet quãlibet aliquo­
                  <lb/>
                tam habet) vt ly quãlibet diſtribuat pro generibꝰ
                  <lb/>
                ſingulorū (et nõ quãlibet habet rationalē ꝓporti-
                  <lb/>
                onē: igitur aliquam habet que eſt irrationalis ꝓ-
                  <lb/>
                portio: et illi eſt cõmenſurabilis / vt patet ex quarta
                  <lb/>
                ſuppoſitione: igitur ꝓpropoſitū. </s>
                <s xml:id="N14CCE" xml:space="preserve">Probat̄̄ antecedēs /
                  <lb/>
                qm̄ inter nulliꝰ ꝓportionis terminos inueniūtur
                  <lb/>
                tot numeri cõtinuo ꝓportionabiles quot poſſunt
                  <lb/>
                ſignari partes aliquote: igitur aliqua pars ali-
                  <lb/>
                quota erit ꝓportio irratiõalis. </s>
                <s xml:id="N14CD9" xml:space="preserve">Et ſic ptꝫ ↄ̨cluſio:</s>
              </p>
              <p xml:id="N14CDC">
                <s xml:id="N14CDD" xml:space="preserve">Quinta cõcluſio. </s>
                <s xml:id="N14CE0" xml:space="preserve">Non oīs proportio
                  <lb/>
                irrationalis eſt ſubdupla, aut ſubtripla, et ſic con­
                  <lb/>
                ſequēter ad aliquã irrationalē: īmo multe irrati-
                  <lb/>
                onales ſunt ſubduple aut ſubtriple .etc̈. ad ratio-
                  <lb/>
                nales. </s>
                <s xml:id="N14CEB" xml:space="preserve">Probatur hec ↄ̨cluſio facile: qm̄ medietas
                  <lb/>
                duple, quintuple, triple, octuple .etc̈. nõ eſt ſubdu-
                  <lb/>
                pla ad aliquã irrationalē: et tñ eſt irrationalis / vt
                  <lb/>
                ſatis patet ex decima ↄ̨cluſione cū ſuo primo cor-
                  <lb/>
                relario precedentis capitis / igitur concluſio vera.</s>
              </p>
              <p xml:id="N14CF6">
                <s xml:id="N14CF7" xml:space="preserve">Sexta concluſio. </s>
                <s xml:id="N14CFA" xml:space="preserve">Quelibet ꝓportio
                  <cb chead="Capitulum ſeptimū."/>
                in qualibet proportione rationali ab aliqua pro­
                  <lb/>
                portione rationali vel irratiõali exceditur. </s>
                <s xml:id="N14D02" xml:space="preserve">Pro-
                  <lb/>
                batur hec concluſio: quoniã data quacū propor­
                  <lb/>
                tione ad illam poteſt dari dupla, tripla, quadru­
                  <lb/>
                pla, et ſic cõſequenter procedendo per oēs ſpecies
                  <lb/>
                ꝓportionis multiplicis: quoniã poſſunt dari tres
                  <lb/>
                termini continuo ꝓportionabiles tali ꝓportione
                  <lb/>
                data: et quatuor, et quin, et ſex, et ſic conſequēter
                  <lb/>
                vt docet ſexta ſuppoſitio: et etiam data quacun
                  <lb/>
                dabitur vna que contineat ipſam et medietatē eiꝰ
                  <lb/>
                et alia que continet ipſam et vnã tertiã eius, et vnã
                  <lb/>
                quartam, et ſic in infinituꝫ. </s>
                <s xml:id="N14D19" xml:space="preserve">Item dabitur vna que
                  <lb/>
                cõtinet ipſam et duas tertias eius, vel tres quar-
                  <lb/>
                tas: et ſic in infinītum ſecundū omnē ſpeciem pro-
                  <lb/>
                portionis rationalis tam ſimplicis quam cõpo-
                  <lb/>
                ſite: et quelibet talis proportio erit rationalis vel
                  <lb/>
                irrationalis / vt patet ex primo capite prime par-
                  <lb/>
                tis: igitur quelibet proportio in qualibet propor­
                  <lb/>
                tione rationali ab aliqua proportione rationali
                  <lb/>
                vel irrationali exceditur. </s>
                <s xml:id="N14D2C" xml:space="preserve">Patet igitur concluſio.</s>
              </p>
              <p xml:id="N14D2F">
                <s xml:id="N14D30" xml:space="preserve">Septima cõcluſio. </s>
                <s xml:id="N14D33" xml:space="preserve">Quelibet ꝓpor-
                  <lb/>
                tio in qualibet proportione rationali aliquã ra-
                  <lb/>
                tionalem vel irratiõalem excedit. </s>
                <s xml:id="N14D3A" xml:space="preserve">Probatur / qm̄
                  <lb/>
                quelibet proportio poteſt diuidi in duas equales
                  <lb/>
                ratiõales vel non rationales: in .3. in .4. in .5. in .6. /
                  <lb/>
                et ſic in infinitū. </s>
                <s xml:id="N14D43" xml:space="preserve">vt patet ex quinta ſuppoſitione / et
                  <lb/>
                ſui medietatē in proportione dupla excedit: et ter-
                  <lb/>
                tiã in tripla: et quartã in q̈drupla: et ſic in infinitū /
                  <lb/>
                vt patet ex prima ſuppoſitione: et duas tertias in
                  <lb/>
                ſexquialtera: et tres quartas ī ſexquitertia: et tres
                  <lb/>
                quintas in ſuprabipartiente tertias: et ſic in infi-
                  <lb/>
                nitum diſcurrendo per ſingulas ſpecies propor-
                  <lb/>
                tionuꝫ rationalium: igitur quelibet proportio in
                  <lb/>
                qualibet proportione rationali aliquam ratio-
                  <lb/>
                nalem vel irrationalem excedit.</s>
              </p>
              <p xml:id="N14D58">
                <s xml:id="N14D59" xml:space="preserve">Ad generandas autē proportiones
                  <lb/>
                irrationales inter terminos proportionis ratio­
                  <lb/>
                nalis mediantes ſit.</s>
              </p>
              <p xml:id="N14D60">
                <s xml:id="N14D61" xml:space="preserve">Octaua cõcluſio que vocat̄̄ cõcluſio
                  <lb/>
                medie rei inuentionis. </s>
                <s xml:id="N14D66" xml:space="preserve">Si datis duabus rectis li-
                  <lb/>
                neis proportionabilibus proportione rationali
                  <lb/>
                vel irrationali in directum protractis coniūctis
                  <lb/>
                at ligatis: deſcribatur ſemicirculus: et a cõmuni
                  <lb/>
                medio ſiue puncto in quo vniuntur eleuetur linea
                  <lb/>
                directe orthogonaliter ad peripheriam vſ ſemi­
                  <lb/>
                circuli. </s>
                <s xml:id="N14D75" xml:space="preserve">talis linea ſcḋm cõtinuã ꝓportionalitatē
                  <lb/>
                inter datas lineas mediabit. </s>
                <s xml:id="N14D7A" xml:space="preserve">Huiꝰ cõcluſionis ſen­
                  <lb/>
                ſus talis eſt. </s>
                <s xml:id="N14D7F" xml:space="preserve">Si velis inter duas lineas ꝓportiõa-
                  <lb/>
                biles ꝓportione dupla aut quacun alia īuenire
                  <lb/>
                vnã que ſe habeat in eadē ꝓportione ad minorē in
                  <lb/>
                qua ſe habet maior ad ipſam: ↄ̨iūge illas duas li­
                  <lb/>
                neas et ſuꝑ illas deſcribas ſemicirculū: et a pūcto
                  <lb/>
                in quo iūgunt̄̄ ille due linee oriat̄̄ directe et ortho-
                  <lb/>
                gonaliter vna alia linea vſ ad circūferentiã cir-
                  <lb/>
                culi: et illa eſt linea q̄ querit̄̄: et ꝓportio maioris li-
                  <lb/>
                nee ad illã mediã eſt medietas ꝓportiõis q̄ eſt īter
                  <lb/>
                illã lineã maiorē et minimã ſic ↄ̨iunctas. </s>
                <s xml:id="N14D94" xml:space="preserve">Exemplū /
                  <lb/>
                huius concluſionis patet in hac figura.</s>
              </p>
              <figure xml:id="N14D99" number="5">
                <image file="0049-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YHKVZ7B4/figures/0049-01"/>
              </figure>
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