Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N13CDA" level="3" n="5" type="chapter" type-free="capitulum">
              <pb chead="Secunde partis" file="0040" n="40"/>
              <p xml:id="N13D28">
                <s xml:id="N13D29" xml:space="preserve">Quarta ſuppoſitio ſiue petitio. </s>
                <s xml:id="N13D2C" xml:space="preserve">Oē
                  <lb/>
                quod ſucceſſiue diminuitur vſ ad non gradū eſt
                  <lb/>
                latitudo diuiſibilis: et in duas medietates: et ī tres
                  <lb/>
                tertias, et in quatuor quartas, et ſic conſequenter
                  <lb/>
                </s>
                <s xml:id="N13D36" xml:space="preserve">Diminuitur enim ad ſubduplum, ad ſubtriplum,
                  <lb/>
                ad ſubquadruplum: et ſic deinceps.</s>
              </p>
              <p xml:id="N13D3B">
                <s xml:id="N13D3C" xml:space="preserve">Quīta ſuppoſitio. </s>
                <s xml:id="N13D3F" xml:space="preserve">Latitudo propor-
                  <lb/>
                tionis maiores inequalitatis eſt ſucceſſiue dimi-
                  <lb/>
                nuibilis vſ ad nõ gradum. </s>
                <s xml:id="N13D46" xml:space="preserve">Probatur tum pri-
                  <lb/>
                mo / quia maius extremum proportionis maioris
                  <lb/>
                īequalitatis ſucceſſiue valet diminui vſ ad equa­
                  <lb/>
                litatē minoris extremi: et in tali diminutione pro-
                  <lb/>
                portio maioris inequalitatis ſucceſſiue diminui-
                  <lb/>
                tur ad non gradum / vt conſtat: igitur in tali di-
                  <lb/>
                minutione quelibet proportio minor illa ſignata
                  <lb/>
                dabitur. </s>
                <s xml:id="N13D57" xml:space="preserve">Tum ſecundo / q2 vt baſanus concedit ve­
                  <lb/>
                locitas motus correſpondet magnitudini ꝓpor-
                  <lb/>
                tionis quo ad equalitatē: ſed ipſa velocitas mo-
                  <lb/>
                tus eſt diminuibilis continuo ſucceſſiue vſ ad nõ
                  <lb/>
                gradū: igitur et latitudo proportionis ſibi corre-
                  <lb/>
                ſpondens inequalitate. </s>
                <s xml:id="N13D64" xml:space="preserve">¶ Ex hac ſequitur /  que-
                  <lb/>
                libet latitudo proportionis maioris inequalita-
                  <lb/>
                tis diuidi poteſt in duas medietates, in tres ter-
                  <lb/>
                tias, in quatuor quartas, et ſic deinceps. </s>
                <s xml:id="N13D6D" xml:space="preserve">Patet
                  <lb/>
                hoc correlariū ex priore auxiliante quarta.</s>
              </p>
              <p xml:id="N13D72">
                <s xml:id="N13D73" xml:space="preserve">Sexta ſuppoſitio. </s>
                <s xml:id="N13D76" xml:space="preserve">Omne quod effi-
                  <lb/>
                citur ſubduplū ad id quod erat antea perdit me-
                  <lb/>
                dietatem ſui: et id quod remanet eſt tantū quantuꝫ
                  <lb/>
                eſt id quod perdidit / qm̄ perdidit aliã medietatem
                  <lb/>
                et cuiuſlibet quanti medietates ſunt equales.</s>
              </p>
              <p xml:id="N13D81">
                <s xml:id="N13D82" xml:space="preserve">His ſuppoſitis aduertendū eſt /  ba-
                  <lb/>
                ſanus volens defenſare quãlibet proportionalē ra-
                  <lb/>
                tionalē cuilibet alteri eſſe cõmenſurabilē aſtruxit
                  <lb/>
                proportionū cõmenſurabilitatē ſiue ꝓportioneꝫ
                  <lb/>
                aſſumendã eſſe ex denominationū ꝓportionibus
                  <lb/>
                ponens talem concluſionē. </s>
                <s xml:id="N13D8F" xml:space="preserve">Proportionū propor­
                  <lb/>
                tio eſt earū denominationū proportio: vt quadru­
                  <lb/>
                pla eſt dupla ad duplã: q2 inter earum denomina­
                  <lb/>
                tiones ſiue numeros a quibus denominantur eſt
                  <lb/>
                proportio dupla, a binario enim dupla: et a qua-
                  <lb/>
                ternario quadrupla denomīatur. </s>
                <s xml:id="N13D9C" xml:space="preserve">Item dupla eſt
                  <lb/>
                ſexquitertia ad ſexquialteram: q2 dupla a bina-
                  <lb/>
                rio ſexquialtera vero ab vnitate cū dimidio deno­
                  <lb/>
                minatur. </s>
                <s xml:id="N13DA5" xml:space="preserve">Conſtat autem binarii ad vnitatem cum
                  <lb/>
                dimidio proportionem ſexquitertiam eſſe.</s>
              </p>
              <note position="left" xml:id="N13DAA" xml:space="preserve">Contra
                <lb/>
              baſanū
                <lb/>
              primo.</note>
              <p xml:id="N13DB2">
                <s xml:id="N13DB3" xml:space="preserve">Sed contra hanc opinationem mea
                  <lb/>
                ſententia mathemathicis principiis derogantē et
                  <lb/>
                contrariã: arguitur primo ſic. </s>
                <s xml:id="N13DBA" xml:space="preserve">Ex hac opinione ſe­
                  <lb/>
                quitur octuplam eſſe duplaꝫ ad quadruplam: ſed
                  <lb/>
                conſequens eſt manifeſte falſū: igitur illud ex quo
                  <lb/>
                ſequitur. </s>
                <s xml:id="N13DC3" xml:space="preserve">Sequela probatur / q2 illarū proportio-
                  <lb/>
                onū octuple videlicet et quadruple denominatio-
                  <lb/>
                nes ſiue numeros a quibus denominãtur, duple
                  <lb/>
                ꝓportiõis rationē habere conſtat .8. em̄ ad .4. du­
                  <lb/>
                pla ꝓportio eſt: igitur expoſitiõe octupla dupla
                  <lb/>
                eſt ad quadruplã. </s>
                <s xml:id="N13DD0" xml:space="preserve">Iã falſitatē cõſequentis oſten-
                  <lb/>
                damus ſuꝑeſt: qm̄ ſi octupla eſt dupla ad quadru-
                  <lb/>
                plã: ſequitur /  quadrupla eſt medietas ipſiꝰ octu­
                  <lb/>
                ple: vt ptꝫ ex prima ſuppoſitione: ſed cõſequēs eſt
                  <lb/>
                falſum: igitur illud ex quo ſequitur: q2 tūc ſeque-
                  <lb/>
                retur /  octupla cõtineret quadruplã bis adequa­
                  <lb/>
                te: ſed hoc eſt falſuꝫ / q2 cõtinet quadruplã et duplã
                  <lb/>
                adequate / vt ptꝫ in his terminis .8. ad .4. et .4. ad .1.
                  <lb/>
                </s>
                <s xml:id="N13DE2" xml:space="preserve">Patet hec conſequentia ex ſecunda parte eiuſdē
                  <lb/>
                ſuppoſitionis.
                  <note position="left" xlink:href="note-0040-01a" xlink:label="note-0040-01" xml:id="N13E6D" xml:space="preserve">Cõfirma­
                    <lb/>
                  tio ṗma.</note>
                </s>
                <s xml:id="N13DEC" xml:space="preserve">¶ Et confirmatur / q2 omne dupluꝫ
                  <lb/>
                ad aliquod continet ipſum vel equale ei bis tantū
                  <cb chead="Capitulū quintū."/>
                ſed octupla eſt dupla ad quadruplã per te / igitur
                  <lb/>
                continet ipſum bis tantū: ſed cõſequens eſt falſuꝫ /
                  <lb/>
                q2 ſexdecupla cõtinet quadruplã bis tantū. </s>
                <s xml:id="N13DF8" xml:space="preserve">Cõſe-
                  <lb/>
                quentia ptꝫ ex ſe: et minor eſt prima pars ſecunde
                  <lb/>
                ſuppoſitionis.
                  <note position="right" xlink:href="note-0040-02a" xlink:label="note-0040-02" xml:id="N13E75" xml:space="preserve">Cõfirma­
                    <lb/>
                  tio ſcḋa</note>
                </s>
                <s xml:id="N13E04" xml:space="preserve">¶ Confirmatur ſecundo / q2 ſi poſi-
                  <lb/>
                tio eſſet vera / ſequeretur /  dupla eſſet medietas
                  <lb/>
                octuple: ſed hoc eſt falſum: igitur illud ex quo ſe-
                  <lb/>
                quitur: q2 ſecundū iſtã opinionē octupla eſt qua-
                  <lb/>
                drupla ad duplã / vt ptꝫ ex ꝓportione denoīationū
                  <lb/>
                duple et octuple: et ſi octupla eſt quadrupla ad du­
                  <lb/>
                plã iam ſequitur /  ipſa dupla eſt quarta octuple
                  <lb/>
                et nõ medietas. </s>
                <s xml:id="N13E15" xml:space="preserve">Quodlibet em̄ eſt quadruplū ad
                  <lb/>
                ſui quartã: cum eã contineat quater adequate. </s>
                <s xml:id="N13E1A" xml:space="preserve">Iã
                  <lb/>
                probatur ſequela: et capio ꝓportionē octuplam: et
                  <lb/>
                volo /  diminuatur quouſ fiat quadrupla ade-
                  <lb/>
                quate: vt poſito /  octo diminuãtur vſ ad quatu­
                  <lb/>
                or: et arguitur ſic: ipſa proportio octupla efficitur
                  <lb/>
                in duplo minor vĺ cõcedit poſitio. </s>
                <s xml:id="N13E27" xml:space="preserve">Efficitur enim
                  <lb/>
                q̈drupla que eſt ſubdupla ad octuplã: igitur ipſa
                  <lb/>
                proportio octupla perdit adequate medietatem
                  <lb/>
                ſui / vt ptꝫ ex tertia ſuppoſitione: et non perdit niſi
                  <lb/>
                duplã adequate / vt conſtat / igitur dupla eſt medie­
                  <lb/>
                tas octuple / qḋ fuit inferendū.
                  <note position="right" xlink:href="note-0040-03a" xlink:label="note-0040-03" xml:id="N13E7D" xml:space="preserve">3. confir-
                    <lb/>
                  matio</note>
                </s>
                <s xml:id="N13E39" xml:space="preserve">¶ Et ↄ̨firmat̄̄ tertio /
                  <lb/>
                q2 ſi iſta poſitio eſſet vera / ſeq̄ret̄̄ /  dupla eſſet eq̈-
                  <lb/>
                lis quadruple. </s>
                <s xml:id="N13E40" xml:space="preserve">Cõſequēs eſt falſum et cõtra opi-
                  <lb/>
                nantem / igitur illud ex quo ſequitur. </s>
                <s xml:id="N13E45" xml:space="preserve">Sequela ar-
                  <lb/>
                guitur / et volo /  potentia vt octo moueat reſiſten­
                  <lb/>
                tiam vt vnum velocitate vt quatuor exempli gra-
                  <lb/>
                tia / deinde volo /  potētia ſtante reſiſtentia: dimi-
                  <lb/>
                nuatur vſ ad ſubduplū: et arguo ſic / ille motꝰ ſiue
                  <lb/>
                velocitas vt quatuor diminuetur ad ſubduplum:
                  <lb/>
                igitur perdit medietatē ſui. </s>
                <s xml:id="N13E54" xml:space="preserve">Patet cõſequentia ex
                  <lb/>
                ſuppoſitione tertia: et per cõſequens nõ manebit
                  <lb/>
                niſi velocitas vt duo: et deperdet̄̄ velocitas vt duo /
                  <lb/>
                igitur tanta ꝓportio deperdita eſt quanta manet
                  <lb/>
                </s>
                <s xml:id="N13E5E" xml:space="preserve">Patet hec cõſequētia / q2 ab equalibꝰ ꝓportioni-
                  <lb/>
                bus equales latitudines motuū ꝓueniūt: ſed ma-
                  <lb/>
                net quadrupla / ergo deperdita eſt ei equalis: ſed
                  <lb/>
                deperdita eſt dūtaxat ꝓportio dupla: ergo dupla
                  <lb/>
                eſt equalis quadruple: quod fuit inferendum.</s>
              </p>
              <p xml:id="N13E85">
                <s xml:id="N13E86" xml:space="preserve">Secundo arguitur ſic / ſi illa poſitio
                  <lb/>
                eſſet vera / ſequeretur /  quarta alicuiꝰ et ſua medie­
                  <lb/>
                tas eſſent equales ſed cõſequens eſt falſum: igitur
                  <lb/>
                illud ex quo ſequitur. </s>
                <s xml:id="N13E8F" xml:space="preserve">Sequela ꝓbatur / q2 dupla
                  <lb/>
                eſt quarta pars octuple et medietas octuple ꝑ po-
                  <lb/>
                ſitionē: igitur propoſitū. </s>
                <s xml:id="N13E96" xml:space="preserve">Maior probatur / q2 du­
                  <lb/>
                pla eſt quarta pars ipſius octuple cū octuple ad
                  <lb/>
                duplam ſit proportio quadrupla / vt patet ex po-
                  <lb/>
                ſitiõe. </s>
                <s xml:id="N13E9F" xml:space="preserve">Minor probatur: et volo /  octupla perdat
                  <lb/>
                ꝓportionē duplã adequate: et manifeſtū eſt /  effi-
                  <lb/>
                citur quadrupla: et per cõſequens ſubdupla ad id
                  <lb/>
                quod erat antea / vt patet ex poſitione: igitur ꝑdit
                  <lb/>
                medietatē ſui. </s>
                <s xml:id="N13EAA" xml:space="preserve">Patet cõſequentia ex tertia et ſexta
                  <lb/>
                ſuppoſitionibus: et non perdit niſi duplam: ergo
                  <lb/>
                dupla eſt medietas octuple / quod fuit probandū.
                  <lb/>
                  <note position="right" xlink:href="note-0040-04a" xlink:label="note-0040-04" xml:id="N13EDD" xml:space="preserve">Cõfirma­
                    <lb/>
                  tio ṗma.</note>
                </s>
                <s xml:id="N13EB8" xml:space="preserve">¶ Et confirmatur / quia ſi poſitio eſſet vera / ſeque-
                  <lb/>
                retur /  aliquid contineret alterum bis adequate
                  <lb/>
                et tamen non eſſet duplum ad illud: ſed minꝰ quaꝫ
                  <lb/>
                duplum: conſequens eſt manifeſte falſuꝫ et contra
                  <lb/>
                diffinitionem proportionis duple: igitur. </s>
                <s xml:id="N13EC3" xml:space="preserve">Seque­
                  <lb/>
                la probatur: quia proportio dupla ſexquiquar-
                  <lb/>
                ta bis adequate continet ſexquialteram: patet
                  <lb/>
                in his terminis .9.6.4. </s>
                <s xml:id="N13ECC" xml:space="preserve">Nouem enim ad quatuor
                  <lb/>
                eſt proportio dupla ſexquiquarta: et componitur
                  <lb/>
                adequate ex proportione .9. ad .6. et .6. ad .4. qua­
                  <lb/>
                rum vtra eſt ſexquialtera: et tamen ipſa propor­
                  <lb/>
                tio dupla ſexquiquarta eſt minor quam dupla
                  <lb/>
                ad ſexquialteram: igitur propoſitum.</s>
              </p>
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