Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Secunde partis" file="0034" n="34"/>
                ior ꝓportione ſuꝑparticulari aut ſuprapartiente
                  <lb/>
                </s>
                <s xml:id="N133AA" xml:space="preserve">Conſequētia eſt nota ex tertia ſuppoſitione et an-
                  <lb/>
                tecedēs ꝓbatur: q2 denominationes illaꝝ ꝓpor-
                  <lb/>
                tionum multiplicis, multiplicis ſuꝑparticularis,
                  <lb/>
                et multiplicis ſuprapartientis, ſumūtur a nūero
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                vel numero cum fractione: denominationis vero
                  <lb/>
                ſuꝑparticularis, aut ſuprapartientis, ſumuntur
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                ab vnitate cū fractione: vt patet ex correlariis ſe-
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                cunde ſuppoſitionis huiꝰ capitis: igitur denomi-
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                nationes illaꝝ puta multiplicis: multiplicis .etc̈.
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                ſunt maiores quã ſuꝑparticularis aut ſuprapar-
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                tientis. </s>
                <s xml:id="N133C1" xml:space="preserve">Et ſic patet cõcluſio.
                  <note position="left" xlink:href="note-0034-01a" xlink:label="note-0034-01" xml:id="N13419" xml:space="preserve">1. correla­
                    <lb/>
                  rium.</note>
                </s>
                <s xml:id="N133C9" xml:space="preserve">¶ Ex qua ſequitur pri­
                  <lb/>
                mo:  ꝓportiones multiplices ſuꝑparticulares: et
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                multiplices ſuprapartientes ſunt maiores ꝓpor-
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                tionibꝰ multiplicibꝰ: ita  quelibet multiplex
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                ſuꝑparticĺaris, aut ſuprapartiēs, qualibet mul-
                  <lb/>
                tiplici ab eodē numero denominata eſt maior: vt
                  <lb/>
                dupla ſexquialtera eſt maior dupla: tripla ſexqui­
                  <lb/>
                quarta maior tripla: tripla em̄ et tripla ſexquiq̈r­
                  <lb/>
                ta ab eodē numero denominantur: ſed nõ adequa­
                  <lb/>
                te. </s>
                <s xml:id="N133DE" xml:space="preserve">Patet hoc correlariū eo modo quo concluſio.
                  <lb/>
                  <note position="left" xlink:href="note-0034-02a" xlink:label="note-0034-02" xml:id="N13421" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N133E8" xml:space="preserve">¶ Sequitur ſecūdo:  ex dictis faciliter eſt inueni­
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                re modū cognoſcendi ꝓpoſitis ꝓportiõe ſuꝑpar-
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                ticulari et ſuprapartiēte: que illaꝝ ſit maior. </s>
                <s xml:id="N133EF" xml:space="preserve">Pro­
                  <lb/>
                batur: et ꝓponantur due ꝓportiones a. ſuꝑparti-
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                cularis et b. ſuprapartiēs: et cū quelibet ſuprapar­
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                tiens denominetur ab vnitate cū fratione partiū
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                aliquotaꝝ nõ facientiū vnã: et quelibet ſuꝑparti-
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                cularis ab vnitate cū fractiõe partis aliquote: vt
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                dictū eſt: et omne aggregatū ex partibus aliquotꝪ
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                alicuiꝰ nõ facientibus vnã eſt qualibet parte ali-
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                quota eiuſdē maius vel minꝰ: vel igitur illud ag-
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                gregatū partiū aliquotaꝝ a quo denoīatur ꝓpor­
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                tio b. ſuprapartiens eſt maius parte aliquota a
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                qua denomīatur ꝓportio a. ſuꝑparticularis: aut
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                minus: ſi maius tūc ꝓportio ſuprapartiēs eſt ma-
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                ior data ꝓportione ſuꝑparticulari a. </s>
                <s xml:id="N1340C" xml:space="preserve">Sin minus
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                tunc ꝓportio ſuꝑparticularis eſt maior data ꝓ-
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                portiõe b. ſuprapartiente: qm̄ denomīatur ab vni­
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                tate cū maiori fractione.</s>
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              <p xml:id="N13427">
                <s xml:id="N13428" xml:space="preserve">Secunda concluſio. </s>
                <s xml:id="N1342B" xml:space="preserve">Oīs proportio
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                extremi ad extremū cõponitur ex qualibet minori
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                ꝓportiõe illa: vt ꝓportio dupla cõponitur ex qua­
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                libet ꝓportione ſuprapartiente: et qualibet ſuper­
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                particulari. </s>
                <s xml:id="N13436" xml:space="preserve">Et diſtribuat ly qualibet pro generi-
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                bus ſinguloꝝ. </s>
                <s xml:id="N1343B" xml:space="preserve">Probatur hec cõcluſio oſtenſiue ex
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                quarta ſuppoſitione: qm̄ ſi omne cõpoſitū ex quã­
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                tolibet minori eo cõponitur: et oīs ꝓportio eſt cõ-
                  <lb/>
                poſita ex aliquibus ꝓportionibus / vt ſupponitur
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                cõſequens eſt /  oīs ꝓportio ex qualibet mīori ea
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                cõponatur / quod fuit ꝓbandū.
                  <note position="left" xlink:href="note-0034-03a" xlink:label="note-0034-03" xml:id="N134D2" xml:space="preserve">1: correĺ.</note>
                </s>
                <s xml:id="N1344D" xml:space="preserve">¶ Ex hac cõcluſiõe
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                ſequitur primo:  quelibet ꝓportio cõponitur ex
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                qualibet ꝓportione medioꝝ ad īuicē: et mediorum
                  <lb/>
                ad extrema. </s>
                <s xml:id="N13456" xml:space="preserve">vt ꝓportio dupla que eſt inter .8. et .4.
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                cõponitur ex ꝓportione .7. ad .6. et .6. ad .5. que ſūt
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                ꝓportiones medioꝝ: et ex ꝓportione .8. ad .7. et .5.
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                ad .4. que ſunt extremi ad mediū et medii ad extre­
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                mū. </s>
                <s xml:id="N13461" xml:space="preserve">Probatur correlariū: q2 quelibet talis pro-
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                portio eſt pars illius ꝓportiõis extremi ad extre-
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                mū cū cõponat eã: et eſt minor illa vt patet ex ṗma
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                cõcluſione: igitur cõponitur ex qualibet ꝓportiõe
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                medioꝝ: et medioꝝ ad extrema.
                  <note position="left" xlink:href="note-0034-04a" xlink:label="note-0034-04" xml:id="N134D8" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N13471" xml:space="preserve">¶ Sequitur ſecūdo /
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                 oīs ꝓportio ex infinitis ꝓportionibus cõponit̄̄
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                </s>
                <s xml:id="N13477" xml:space="preserve">Probatur / qm̄ ex qualibet minore ea cõponitur:
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                vt ptꝫ ex cõcluſione: ſed qualibet data infinite ſunt
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                minores: ergo quelibet ex infinitis cõponit̄̄. </s>
                <s xml:id="N1347E" xml:space="preserve">Pro-
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                batur minor / q2 ymaginor quãlibet proportionē
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                inequalitatis eſſe latitudinē in infinitū diuiſibilē
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                q2 alias nõ poſſet augeri nec ad nõ gradū ꝓpor-
                  <cb chead="Capitulum quartū."/>
                tionis inequalitatis ſucceſſiue diminui.
                  <note position="right" xlink:href="note-0034-05a" xlink:label="note-0034-05" xml:id="N134DE" xml:space="preserve">3. correĺ.</note>
                </s>
                <s xml:id="N1348F" xml:space="preserve">¶ Sequit̄̄
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                tertio:  oīs ꝓportio poteſt in infinitas ꝓportio-
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                nes diuidi: que ꝓportiones ſe habebūt vt partes
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                ꝓportionales illiꝰ: et hoc qua volueris ꝓportiõe.
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                </s>
                <s xml:id="N13499" xml:space="preserve">Patet: q2 cū quelibet ꝓportio ſit latitudo quedã:
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                ipſa habet medietatē, tertiã, quartã, ſextam, et ſic
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                deinceps: et ꝑ cõſequens quauis ꝓportione diuiſi­
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                bilis eſt in infinitas ꝓportiones que ſunt partes
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                ꝓportionales eius. </s>
                <s xml:id="N134A4" xml:space="preserve">¶ Sequit̄̄ quarto:  ſi aliqua
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                ꝓportio maioris inequalitatis diminuatur vſ
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                ad ꝓportionē equalitatis neceſſe eſt ipſam conti-
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                nuo ſucceſſiue tranſire per īfinitas ꝓportiones mi­
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                nores ea: vt ſi ꝓportio .8. ad .4. deueniat ad ꝓpor­
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                tioneꝫ equalitatis per diminutionem ipſorum .8.
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                vſ ad .4. neceſſe eſt eã tranſire per oēs ꝓportiões
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                ex quibus cõponitur talis ꝓportio .8. ad .4. et ille
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                ſunt infinte vt dicit ſecundū correlariū: igit̄̄. </s>
                <s xml:id="N134B7" xml:space="preserve">Ma­
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                ior patet / q2 cū cõtinuo aliquid diminuitur vſ ad
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                certã quantitatē per infinitas minores quantita­
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                tes tranſit: vt notū eſt. </s>
                <s xml:id="N134C0" xml:space="preserve">Et ſic ſimiliter eſt de quali-
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                bet latitudine que continuo ſucceſſiue diminuitur
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                ſed ꝓportio .8. ad .4. eſt latitudo que continuo ſuc­
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                ceſſiue diminuitur (vt pono) igitur. </s>
                <s xml:id="N134C9" xml:space="preserve">et ſic patet cor-
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                relariū: qm̄ eo modo ꝓbabis de quauis alia.</s>
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              <p xml:id="N134E4">
                <s xml:id="N134E5" xml:space="preserve">Tertia concluſio. </s>
                <s xml:id="N134E8" xml:space="preserve">Quãlibet propor-
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                tionē in duas equales ꝓportiões ſecare: vt capta
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                ꝓportione que eſt .8. ad .4. ipſa in duas inequales
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                diuidetur inuento numero ſine termino equaliter
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                diſtante ab vtro extremoꝝ: puta īuento numero
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                ſenario .8. em̄ ad .6. eſt ꝓportio ſexquitertia: et .6.
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                ad .4. proportio ſexquialtera: et hec maior eſt illa.
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                </s>
                <s xml:id="N134F8" xml:space="preserve">Probatur hec concluſio: q2 aut talis ꝓportio da­
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                tur inter duas quantitates cõtinuas: aut inter du­
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                os numeros: ſi inter duas quantitates cõtinuas:
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                ille erunt inequales: qm̄ de ꝓportione maioris in­
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                equalitatis loquimur: capiatur igitur quantitas
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                media inter illas que equaliter diſtat ab vtra il­
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                larū: et tunc manifeſtū eſt /  maioris illaꝝ quanti-
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                tatū ad quãtitatē mediã eſt vna ꝓportio: et medie
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                quantitatis ad minimã illaꝝ eſt vna alia ꝓportio
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                et illa ꝓportio que eſt inter illas quantitates di-
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                uiditur in illas duas ꝓportiones ītermedias, q2
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                ex illis cõponitur / vt patet ex primo correlario ſe-
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                cunde concluſionis: et prima illaꝝ que videlicet eſt
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                maioris quantitatis ad mediã minor eſt illa que
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                eſt medie ad alterū extremū minꝰ: igitur talis ꝓ-
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                portio diuiditur in duas proportiões inequales /
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                quod fuit ꝓbandū. </s>
                <s xml:id="N1351B" xml:space="preserve">Minor ꝓbatur: q2 illa quãti-
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                tas media ꝑ tantū excedit minus extremū: ꝑ quan­
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                tū adequate maius extremū excedit illã: igit̄̄ ma-
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                ior eſt ꝓportio illius quantitatis medie ad minus
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                extremū: quã alteriꝰ extremi puta maioris ad me­
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                diã. </s>
                <s xml:id="N13528" xml:space="preserve">Patet hec cõſequentia ex octaua ſuppoſitiõe
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                huiꝰ capitis. </s>
                <s xml:id="N1352D" xml:space="preserve">Sin autē talis ꝓportio eſt inter nu-
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                meros puta inter a. et c. quoꝝ a. eſt maior et c. mīor /
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                vel igit̄̄ illi nūeri ſunt pares: vĺ nõ pares ſi pares
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                manifeſtū eſt /  aggregatū ex eis eſt nūerus par:
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                et ꝑ cõſequens hꝫ medietatē: et illa medietas eſt me­
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                diū inter illos duos numeros a.c. / vt patet ex ṗmo
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                correlario prime cõcluſionis ſecūdi capitis huiꝰ:
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                ſit igitur illud mediū b. / et ſequit̄̄ /  a. ad b. eſt vna
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                ꝓportio: et b. ad c. eſt vna altera: et ex illis cõponit̄̄
                  <lb/>
                ꝓportio a. ad b. / vt ptꝫ ex primo correlario ſecūde
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                cõcluſionis huiꝰ: et prima illaꝝ que videlicet eſt a.
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                ad b. eſt minor quã illa que eſt b. ad .c. / quod ptꝫ vt
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                ſupra: igitur ꝓportio a. ad c. in duas ꝓportiones
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                inequales ſecatur. </s>
                <s xml:id="N1354A" xml:space="preserve">Sin nõ pares creſcat vter il-
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                loꝝ duoꝝ numeroꝝ ad ſuū duplū: et ſequitur /  eq̈­
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                lem ꝓportionē acquirit maior illoꝝ et minor puta </s>
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