Alvarus, Thomas, Liber de triplici motu, 1509

Table of Notes

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                <s xml:id="N13397" xml:space="preserve">
                  <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
                  <lb/>
                vel numero cum fractione: denominationis vero
                  <lb/>
                ſuꝑparticularis, aut ſuprapartientis, ſumuntur
                  <lb/>
                ab vnitate cū fractione: vt patet ex correlariis ſe-
                  <lb/>
                cunde ſuppoſitionis huiꝰ capitis: igitur denomi-
                  <lb/>
                nationes illaꝝ puta multiplicis: multiplicis .etc̈.
                  <lb/>
                ſunt maiores quã ſuꝑparticularis aut ſuprapar-
                  <lb/>
                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
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                  rium.</note>
                </s>
                <s xml:id="N133C9" xml:space="preserve">¶ Ex qua ſequitur pri­
                  <lb/>
                mo:  ꝓportiones multiplices ſuꝑparticulares: et
                  <lb/>
                multiplices ſuprapartientes ſunt maiores ꝓpor-
                  <lb/>
                tionibꝰ multiplicibꝰ: ita  quelibet multiplex
                  <lb/>
                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
                  <lb/>
                re modū cognoſcendi ꝓpoſitis ꝓportiõe ſuꝑpar-
                  <lb/>
                ticulari et ſuprapartiēte: que illaꝝ ſit maior. </s>
                <s xml:id="N133EF" xml:space="preserve">Pro­
                  <lb/>
                batur: et ꝓponantur due ꝓportiones a. ſuꝑparti-
                  <lb/>
                cularis et b. ſuprapartiēs: et cū quelibet ſuprapar­
                  <lb/>
                tiens denominetur ab vnitate cū fratione partiū
                  <lb/>
                aliquotaꝝ nõ facientiū vnã: et quelibet ſuꝑparti-
                  <lb/>
                cularis ab vnitate cū fractiõe partis aliquote: vt
                  <lb/>
                dictū eſt: et omne aggregatū ex partibus aliquotꝪ
                  <lb/>
                alicuiꝰ nõ facientibus vnã eſt qualibet parte ali-
                  <lb/>
                quota eiuſdē maius vel minꝰ: vel igitur illud ag-
                  <lb/>
                gregatū partiū aliquotaꝝ a quo denoīatur ꝓpor­
                  <lb/>
                tio b. ſuprapartiens eſt maius parte aliquota a
                  <lb/>
                qua denomīatur ꝓportio a. ſuꝑparticularis: aut
                  <lb/>
                minus: ſi maius tūc ꝓportio ſuprapartiēs eſt ma-
                  <lb/>
                ior data ꝓportione ſuꝑparticulari a. </s>
                <s xml:id="N1340C" xml:space="preserve">Sin minus
                  <lb/>
                tunc ꝓportio ſuꝑparticularis eſt maior data ꝓ-
                  <lb/>
                portiõe b. ſuprapartiente: qm̄ denomīatur ab vni­
                  <lb/>
                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
                  <lb/>
                extremi ad extremū cõponitur ex qualibet minori
                  <lb/>
                ꝓportiõe illa: vt ꝓportio dupla cõponitur ex qua­
                  <lb/>
                libet ꝓportione ſuprapartiente: et qualibet ſuper­
                  <lb/>
                particulari. </s>
                <s xml:id="N13436" xml:space="preserve">Et diſtribuat ly qualibet pro generi-
                  <lb/>
                bus ſinguloꝝ. </s>
                <s xml:id="N1343B" xml:space="preserve">Probatur hec cõcluſio oſtenſiue ex
                  <lb/>
                quarta ſuppoſitione: qm̄ ſi omne cõpoſitū ex quã­
                  <lb/>
                tolibet minori eo cõponitur: et oīs ꝓportio eſt cõ-
                  <lb/>
                poſita ex aliquibus ꝓportionibus / vt ſupponitur
                  <lb/>
                cõſequens eſt /  oīs ꝓportio ex qualibet mīori ea
                  <lb/>
                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
                  <lb/>
                ſequitur primo:  quelibet ꝓportio cõponitur ex
                  <lb/>
                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.
                  <lb/>
                cõponitur ex ꝓportione .7. ad .6. et .6. ad .5. que ſūt
                  <lb/>
                ꝓportiones medioꝝ: et ex ꝓportione .8. ad .7. et .5.
                  <lb/>
                ad .4. que ſunt extremi ad mediū et medii ad extre­
                  <lb/>
                mū. </s>
                <s xml:id="N13461" xml:space="preserve">Probatur correlariū: q2 quelibet talis pro-
                  <lb/>
                portio eſt pars illius ꝓportiõis extremi ad extre-
                  <lb/>
                mū cū cõponat eã: et eſt minor illa vt patet ex ṗma
                  <lb/>
                cõcluſione: igitur cõponitur ex qualibet ꝓportiõe
                  <lb/>
                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 /
                  <lb/>
                 oīs ꝓportio ex infinitis ꝓportionibus cõponit̄̄
                  <lb/>
                </s>
                <s xml:id="N13477" xml:space="preserve">Probatur / qm̄ ex qualibet minore ea cõponitur:
                  <lb/>
                vt ptꝫ ex cõcluſione: ſed qualibet data infinite ſunt
                  <lb/>
                minores: ergo quelibet ex infinitis cõponit̄̄. </s>
                <s xml:id="N1347E" xml:space="preserve">Pro-
                  <lb/>
                batur minor / q2 ymaginor quãlibet proportionē
                  <lb/>
                inequalitatis eſſe latitudinē in infinitū diuiſibilē
                  <lb/>
                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
                  <lb/>
                tertio:  oīs ꝓportio poteſt in infinitas ꝓportio-
                  <lb/>
                nes diuidi: que ꝓportiones ſe habebūt vt partes
                  <lb/>
                ꝓportionales illiꝰ: et hoc qua volueris ꝓportiõe.
                  <lb/>
                </s>
                <s xml:id="N13499" xml:space="preserve">Patet: q2 cū quelibet ꝓportio ſit latitudo quedã:
                  <lb/>
                ipſa habet medietatē, tertiã, quartã, ſextam, et ſic
                  <lb/>
                deinceps: et ꝑ cõſequens quauis ꝓportione diuiſi­
                  <lb/>
                bilis eſt in infinitas ꝓportiones que ſunt partes
                  <lb/>
                ꝓportionales eius. </s>
                <s xml:id="N134A4" xml:space="preserve">¶ Sequit̄̄ quarto:  ſi aliqua
                  <lb/>
                ꝓportio maioris inequalitatis diminuatur vſ
                  <lb/>
                ad ꝓportionē equalitatis neceſſe eſt ipſam conti-
                  <lb/>
                nuo ſucceſſiue tranſire per īfinitas ꝓportiones mi­
                  <lb/>
                nores ea: vt ſi ꝓportio .8. ad .4. deueniat ad ꝓpor­
                  <lb/>
                tioneꝫ equalitatis per diminutionem ipſorum .8.
                  <lb/>
                vſ ad .4. neceſſe eſt eã tranſire per oēs ꝓportiões
                  <lb/>
                ex quibus cõponitur talis ꝓportio .8. ad .4. et ille
                  <lb/>
                ſunt infinte vt dicit ſecundū correlariū: igit̄̄. </s>
                <s xml:id="N134B7" xml:space="preserve">Ma­
                  <lb/>
                ior patet / q2 cū cõtinuo aliquid diminuitur vſ ad
                  <lb/>
                certã quantitatē per infinitas minores quantita­
                  <lb/>
                tes tranſit: vt notū eſt. </s>
                <s xml:id="N134C0" xml:space="preserve">Et ſic ſimiliter eſt de quali-
                  <lb/>
                bet latitudine que continuo ſucceſſiue diminuitur
                  <lb/>
                ſed ꝓportio .8. ad .4. eſt latitudo que continuo ſuc­
                  <lb/>
                ceſſiue diminuitur (vt pono) igitur. </s>
                <s xml:id="N134C9" xml:space="preserve">et ſic patet cor-
                  <lb/>
                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-
                  <lb/>
                tionē in duas equales ꝓportiões ſecare: vt capta
                  <lb/>
                ꝓportione que eſt .8. ad .4. ipſa in duas inequales
                  <lb/>
                diuidetur inuento numero ſine termino equaliter
                  <lb/>
                diſtante ab vtro extremoꝝ: puta īuento numero
                  <lb/>
                ſenario .8. em̄ ad .6. eſt ꝓportio ſexquitertia: et .6.
                  <lb/>
                ad .4. proportio ſexquialtera: et hec maior eſt illa.
                  <lb/>
                </s>
                <s xml:id="N134F8" xml:space="preserve">Probatur hec concluſio: q2 aut talis ꝓportio da­
                  <lb/>
                tur inter duas quantitates cõtinuas: aut inter du­
                  <lb/>
                os numeros: ſi inter duas quantitates cõtinuas:
                  <lb/>
                ille erunt inequales: qm̄ de ꝓportione maioris in­
                  <lb/>
                equalitatis loquimur: capiatur igitur quantitas
                  <lb/>
                media inter illas que equaliter diſtat ab vtra il­
                  <lb/>
                larū: et tunc manifeſtū eſt /  maioris illaꝝ quanti-
                  <lb/>
                tatū ad quãtitatē mediã eſt vna ꝓportio: et medie
                  <lb/>
                quantitatis ad minimã illaꝝ eſt vna alia ꝓportio
                  <lb/>
                et illa ꝓportio que eſt inter illas quantitates di-
                  <lb/>
                uiditur in illas duas ꝓportiones ītermedias, q2
                  <lb/>
                ex illis cõponitur / vt patet ex primo correlario ſe-
                  <lb/>
                cunde concluſionis: et prima illaꝝ que videlicet eſt
                  <lb/>
                maioris quantitatis ad mediã minor eſt illa que
                  <lb/>
                eſt medie ad alterū extremū minꝰ: igitur talis ꝓ-
                  <lb/>
                portio diuiditur in duas proportiões inequales /
                  <lb/>
                quod fuit ꝓbandū. </s>
                <s xml:id="N1351B" xml:space="preserve">Minor ꝓbatur: q2 illa quãti-
                  <lb/>
                tas media ꝑ tantū excedit minus extremū: ꝑ quan­
                  <lb/>
                tū adequate maius extremū excedit illã: igit̄̄ ma-
                  <lb/>
                ior eſt ꝓportio illius quantitatis medie ad minus
                  <lb/>
                extremū: quã alteriꝰ extremi puta maioris ad me­
                  <lb/>
                diã. </s>
                <s xml:id="N13528" xml:space="preserve">Patet hec cõſequentia ex octaua ſuppoſitiõe
                  <lb/>
                huiꝰ capitis. </s>
                <s xml:id="N1352D" xml:space="preserve">Sin autē talis ꝓportio eſt inter nu-
                  <lb/>
                meros puta inter a. et c. quoꝝ a. eſt maior et c. mīor /
                  <lb/>
                vel igit̄̄ illi nūeri ſunt pares: vĺ nõ pares ſi pares
                  <lb/>
                manifeſtū eſt /  aggregatū ex eis eſt nūerus par:
                  <lb/>
                et ꝑ cõſequens hꝫ medietatē: et illa medietas eſt me­
                  <lb/>
                diū inter illos duos numeros a.c. / vt patet ex ṗmo
                  <lb/>
                correlario prime cõcluſionis ſecūdi capitis huiꝰ:
                  <lb/>
                ſit igitur illud mediū b. / et ſequit̄̄ /  a. ad b. eſt vna
                  <lb/>
                ꝓportio: et b. ad c. eſt vna altera: et ex illis cõponit̄̄
                  <lb/>
                ꝓportio a. ad b. / vt ptꝫ ex primo correlario ſecūde
                  <lb/>
                cõcluſionis huiꝰ: et prima illaꝝ que videlicet eſt a.
                  <lb/>
                ad b. eſt minor quã illa que eſt b. ad .c. / quod ptꝫ vt
                  <lb/>
                ſupra: igitur ꝓportio a. ad c. in duas ꝓportiones
                  <lb/>
                inequales ſecatur. </s>
                <s xml:id="N1354A" xml:space="preserve">Sin nõ pares creſcat vter il-
                  <lb/>
                loꝝ duoꝝ numeroꝝ ad ſuū duplū: et ſequitur /  eq̈­
                  <lb/>
                lem ꝓportionē acquirit maior illoꝝ et minor puta </s>
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