Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N140E1" level="3" n="6" type="chapter" type-free="capitulum">
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                  <pb chead="Secūde partis" file="0045" n="45"/>
                nalem intermediam: et per conſequens iam nõ ha­
                  <lb/>
                bet ſubquadruplam rationalem. </s>
                <s xml:id="N145C9" xml:space="preserve">Patet hec con-
                  <lb/>
                ſequentia / quia ex oppoſito ſequitur oppoſituꝫ / vt
                  <lb/>
                patet ex decima diffinitione quinti elementorum
                  <lb/>
                euclidis. </s>
                <s xml:id="N145D2" xml:space="preserve">Iam probo priorem conſequentiam vi-
                  <lb/>
                delicet /  ſi inter terminos date proportionis non
                  <lb/>
                fuerit numerus qui ſit medium proportionabile:
                  <lb/>
                non reperiuntur ibi .5. numeri cõtinuo proportio­
                  <lb/>
                nabiles. </s>
                <s xml:id="N145DD" xml:space="preserve">Que probatur ſic: q2 ex oppoſito conſe-
                  <lb/>
                quentis ſequitur oppoſitum ãtecedentis: q2 ſi ſūt
                  <lb/>
                ibi quin numeri continuo ꝓportionabiles iam
                  <lb/>
                ibi tertius numerus eſt medio loco ꝓportionabi-
                  <lb/>
                lis: quia primi ad ipſum eſt ea proportio que ē ip­
                  <lb/>
                ſius ad quintum / vt conſtat: quia ex equalibus cõ-
                  <lb/>
                ponuntur ille ꝓportiones adequate. </s>
                <s xml:id="N145EC" xml:space="preserve">Et ſic proba­
                  <lb/>
                bis alias partes.
                  <note position="left" xlink:href="note-0045-01a" xlink:label="note-0045-01" xml:id="N1461A" xml:space="preserve">correĺm.</note>
                </s>
                <s xml:id="N145F6" xml:space="preserve">¶ Ex hac concluſione ſequitur / 
                  <lb/>
                ſi inter terminos alicuius proportionis fuerit nu­
                  <lb/>
                merus qui ſit medium proportionabile ipſa ha-
                  <lb/>
                bet ſubduplam rationalem et ſi ipſius numeri me­
                  <lb/>
                dii proportio ad aliud extremuꝫ minus date pro-
                  <lb/>
                portionis haberit numerum qui ſit medium pro-
                  <lb/>
                portionabile: tunc tota proportio habet ſubqua­
                  <lb/>
                druplam rationalem: et ſi iteruꝫ illius numeri me­
                  <lb/>
                dii proportio ad minus extremum date ꝓportio-
                  <lb/>
                nis habuerit numerum qui ſit medium ꝓportio-
                  <lb/>
                nabile: iam data proportio habebit ſuboctuplaꝫ
                  <lb/>
                rationalem / et ſic in infinitum. </s>
                <s xml:id="N1460F" xml:space="preserve">Patet hoc correla­
                  <lb/>
                rium ex concluſione et eius ꝓbatione: auxilianti-
                  <lb/>
                bus correlariis ſexte concluſionis ſecūdi capitis</s>
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              <p xml:id="N14620">
                <s xml:id="N14621" xml:space="preserve">Decima concluſio notanda. </s>
                <s xml:id="N14624" xml:space="preserve">Propo­
                  <lb/>
                ſita quauis proportione rationali an habeat ſub­
                  <lb/>
                duplam rationalem inueſtigare. </s>
                <s xml:id="N1462B" xml:space="preserve">vt propoſita du­
                  <lb/>
                pla aut tripla volo īueſtigare et ſcire ex predictis
                  <lb/>
                an habeat ſubduplã rationalem. </s>
                <s xml:id="N14632" xml:space="preserve">Sit propoſita
                  <lb/>
                proportio rationalis f. inter a. numerū maiorem
                  <lb/>
                et b. numerum minoreꝫ. </s>
                <s xml:id="N14639" xml:space="preserve">et volo inueſtigare vtrum
                  <lb/>
                f. ꝓportio habeat ſubduplã rationalem: tunc du-
                  <lb/>
                cam maiorem numerum in minorem / hoc eſt multi­
                  <lb/>
                plicabo a. per b. et ſi numerus inde ꝓueniens fue-
                  <lb/>
                rit quadratus: dico /  habet ſubduplam rationa­
                  <lb/>
                lem. </s>
                <s xml:id="N14646" xml:space="preserve">ſin minus non habet ſubduplam rationalem
                  <lb/>
                </s>
                <s xml:id="N1464A" xml:space="preserve">Probatur prima pars videlicet /  ſi numerus qui
                  <lb/>
                fit ex ductu ipſius a. in b. ſit quadratus: tunc ha-
                  <lb/>
                bet ſubduplam rationalem. </s>
                <s xml:id="N14651" xml:space="preserve">quia ſit talis numerꝰ
                  <lb/>
                eſt quadratus: tunc inter a. et b. eſt medius nume-
                  <lb/>
                rus proportionabilis / vt patet ex quarto correla­
                  <lb/>
                rio ſexte concluſionis ſecundi capitis huius par-
                  <lb/>
                tis: et ſi ſit numerus qui ſit medium ꝓportionabi­
                  <lb/>
                le inter a. et b. / ſequitur /  illa proportio habet ſub­
                  <lb/>
                duplam rationalem. </s>
                <s xml:id="N14660" xml:space="preserve">Patet conſequentia ex cor-
                  <lb/>
                relario precedentis. </s>
                <s xml:id="N14665" xml:space="preserve">Iam probatur ſecunda pars /
                  <lb/>
                quia ſi numerus qui fit ex ductu a. in b. non ſit qua­
                  <lb/>
                dratus: iam inter a. et b. non eſt numerus qui ē me­
                  <lb/>
                dio loco proportionabilis / vt patet ex ſecundo cor­
                  <lb/>
                relario ſexte concluſionis ſecundi capitis huius:
                  <lb/>
                et ſi non eſt numerus qui eſt medio loco proportio­
                  <lb/>
                nabilis inter a. et b. iam ille non habet ſubduplaꝫ
                  <lb/>
                rationalem / vt patet ex concluſione nona huius.</s>
              </p>
              <p xml:id="N14676">
                <s xml:id="N14677" xml:space="preserve">Patet igitur concluſio.
                  <note position="left" xlink:href="note-0045-02a" xlink:label="note-0045-02" xml:id="N14827" xml:space="preserve">correĺm.</note>
                </s>
                <s xml:id="N1467F" xml:space="preserve">¶ Ex hac ſequitur /  du-
                  <lb/>
                pla non habet ſubduplam rationalem, nec tripla
                  <lb/>
                nec octupla, nec aliqua ſuperparticularis. </s>
                <s xml:id="N14686" xml:space="preserve">Pro-
                  <lb/>
                batur / quoniam ducendo quatuor per duo reſul-
                  <lb/>
                tat numerus octonarius qui non eſt quadratus / vt
                  <lb/>
                conſtat: et ducendo .6. per duo: reſultat numerus
                  <lb/>
                duodenarius qui etiam non eſt quadratus: et du­
                  <lb/>
                cendo .16. per duo conſurgit numerus .32. qui non
                  <lb/>
                eſt quadratus vt apparet intelligenti. </s>
                <s xml:id="N14695" xml:space="preserve">Item ducē­
                  <lb/>
                do .3: per duo producuntur .6. qui non ſunt nume-
                  <lb/>
                rus quadratus: et ſic probabis de qualibet alia ꝓ­
                  <cb chead="Capitulum ſextum"/>
                portione ſuperparticulari.
                  <note position="right" xlink:href="note-0045-03a" xlink:label="note-0045-03" xml:id="N1482D" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N146A4" xml:space="preserve">¶ Sequitur ſecundo /
                  <lb/>
                 propoſita qua volueris ꝓportione rationali. </s>
                <s xml:id="N146A9" xml:space="preserve">ī­
                  <lb/>
                ueſtigare poterimus vtrum habeat ſubquadru-
                  <lb/>
                plam rationalē ſuboctuplaꝫ, ſubſexdecuplam, et
                  <lb/>
                ſic in infinitum procedendo per numeros pariter
                  <lb/>
                pares. </s>
                <s xml:id="N146B4" xml:space="preserve">vt propoſita proportione ſexdecupla: vo-
                  <lb/>
                lo inueſtigare: vtrum habeat ſubquadruplam ra­
                  <lb/>
                tionalem, ſuboctuplam, ſubſexdecuplam, et ſic in
                  <lb/>
                infinitum. </s>
                <s xml:id="N146BD" xml:space="preserve">Ad quod inueſtigandum ſiue ſciendum
                  <lb/>
                ſit f. ꝓportio inter a. maiorem numerum et b. mi-
                  <lb/>
                norem: tunc aut inter a. et b. eſt numerus qui ſit me­
                  <lb/>
                dium ꝓportionabile aut non. </s>
                <s xml:id="N146C6" xml:space="preserve">ſi nõ: iam ſequitur /
                  <lb/>
                 non habet ſubquadruplam rationalē nec ſub-
                  <lb/>
                octuplam etc. / vt patet ex nona concluuſione: ſi ſic
                  <lb/>
                ſignetur ille et ſit h. / et tunc videndum eſt an nume­
                  <lb/>
                rus / qui fit ex ductu h. in b. ſit quadratus: et ſi ſic iã
                  <lb/>
                talis ꝓportio f. que eſt inter a. et b. habet ſubqua-
                  <lb/>
                druplam: ſi vero talis numerus non ſit quadratꝰ
                  <lb/>
                dico /  talis proportio non habet ſubquadruplã
                  <lb/>
                rationalem. </s>
                <s xml:id="N146D9" xml:space="preserve">Primum iſtorum probatur. </s>
                <s xml:id="N146DC" xml:space="preserve">quia ſi
                  <lb/>
                talis numerus qui fit ex ductu h. in b. ſit quadra-
                  <lb/>
                tus: iam inter h. et b. eſt numerus medio loco pro-
                  <lb/>
                portionabilis qui ſit k. / vt patet ex quarto correla­
                  <lb/>
                rio preallegato ſexte concluſionis ſecundi capitis
                  <lb/>
                huius: et ex conſequenti iam ꝓportio h. ad b. que
                  <lb/>
                eſt ſubdupla ad ꝓportionem f. habet ſubduplam
                  <lb/>
                proportionem rationalem / vt patet ex correlario
                  <lb/>
                none concluſionis: et ſi habet ſubduplam iam pro­
                  <lb/>
                portio f. habet ſubquadruplam: quia omne ſub-
                  <lb/>
                duplum ſubdupli eſt ſubquadruplum dupli / vt pa­
                  <lb/>
                tet ex ſecundo correlario quarte concluſionis q̈r-
                  <lb/>
                ti capitis huius / quod erat oſtendendum. </s>
                <s xml:id="N146F7" xml:space="preserve">Iam pro­
                  <lb/>
                batur ſecundum: quia ſi numerus qui fit ex ductu
                  <lb/>
                h. in b. non ſit quadratus iam proportio que eſt ī-
                  <lb/>
                ter h. et b. non habet numerū medio loco ꝓportio­
                  <lb/>
                nabilem / vt patet ex ſecundo correlario ſexte con-
                  <lb/>
                cluſionis preallegate: et ſi non habet mediū nume­
                  <lb/>
                rū ꝓportionabilem iã non habet ſubduplã ratio­
                  <lb/>
                nalem: et ſic eius medietas non eſt proportio rõa-
                  <lb/>
                lis et eius medietas eſt ſubquadruplum ꝓportio­
                  <lb/>
                nis f. que eſt a. ad b. / vt cõſtat: igitur proportio ſub­
                  <lb/>
                quadrupla ad f. non eſt rationalis / quod fuit oſtē-
                  <lb/>
                dendum. </s>
                <s xml:id="N14710" xml:space="preserve">Alie particule correlarii ſimilem demon­
                  <lb/>
                ſtrationem ſortiuntur. </s>
                <s xml:id="N14715" xml:space="preserve">Si eni3 non inueniatur ra­
                  <lb/>
                tionalis ſubquadrupla: nec ſuboctuplã rõnalem
                  <lb/>
                inuenies. </s>
                <s xml:id="N1471C" xml:space="preserve">Si vero ſubquadrupla reperta fuerit ra­
                  <lb/>
                tionalis: conſidera an ex ductu vnius extremita-
                  <lb/>
                lis ſubquadrupli in alterum reſultat numerꝰ qua­
                  <lb/>
                dratus: et ſi ſic concludas datam ꝓportionem ha­
                  <lb/>
                bere ſuboctuplam rationalē: quia ſua quarta ha­
                  <lb/>
                bet ſubduplam rationalem. </s>
                <s xml:id="N14729" xml:space="preserve">ſin minus concludas
                  <lb/>
                eam non habere talem ſuboctuplam rationalem.
                  <lb/>
                </s>
                <s xml:id="N1472F" xml:space="preserve">Et ſic in aliis operaberis.
                  <note position="right" xlink:href="note-0045-04a" xlink:label="note-0045-04" xml:id="N14833" xml:space="preserve">3. correl.</note>
                </s>
                <s xml:id="N14737" xml:space="preserve">¶ Sequitur tertio /  ſi­
                  <lb/>
                gnata quauis ꝓportione rationali: inueſtigare et
                  <lb/>
                ſcire poterimus an habeat ſexquialteram ratio-
                  <lb/>
                nalem, ſexquiquartaꝫ, ſexquioctauam, ſexquiſex­
                  <lb/>
                decimã, ſexquitrigeſimã ſecundam, ſexquitrigeſi­
                  <lb/>
                mã quartã, et ſic in infinituꝫ: ꝓcedendo per ſpecies
                  <lb/>
                ꝓportionis ſuperparticularis denominatas a ꝑ­
                  <lb/>
                tibus aliquotis que partes aliquote a nūeris pa-
                  <lb/>
                riter paribus denominantur. </s>
                <s xml:id="N1474A" xml:space="preserve">vt ꝓpoſita ꝓportio­
                  <lb/>
                ne quadrupla: volo inueſtigare et ſcire an ip̄a ha­
                  <lb/>
                beat ſexquialteram rationalem: tūc videbo an ha­
                  <lb/>
                beat medietatem rationalem per doctrinam deci­
                  <lb/>
                me concluſionis huius: et tunc ſi habeat medieta-
                  <lb/>
                tem rationalem: manifeſtum eſt  habet ſexquial<lb/>teram rationalem: quia non oportet ad dandam
                  <lb/>
                ſexquialteram ipſius quadruple aliud quam ad-
                  <lb/>
                dere ipſi quadruple ſuã medietatem puta duplã: </s>
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