Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N12F13" level="3" n="3" type="chapter" type-free="capitulum">
              <p xml:id="N12F21">
                <s xml:id="N12FD0" xml:space="preserve">
                  <pb chead="Secunde partis" file="0032" n="32"/>
                nalis illatio. </s>
                <s xml:id="N12FDC" xml:space="preserve">iſto modo arguendo ſicut ſe ha-
                  <lb/>
                bent octo ad quatuor ita duo ad vnū. </s>
                <s xml:id="N12FE1" xml:space="preserve">igitur ſicut
                  <lb/>
                ſe habēt vnū et duo ad duo ita quatuor et octo ad
                  <lb/>
                octo. </s>
                <s xml:id="N12FE8" xml:space="preserve">Et differt iſte modus arguendi a tertio / quia
                  <lb/>
                in conſequente tertii inferuntur ꝓportiones ma-
                  <lb/>
                ioris inequalitatis in iſto autem inferuntur ꝓpor­
                  <lb/>
                tiones minoris inequalitatis.
                  <note position="left" xlink:href="note-0032-01a" xlink:label="note-0032-01" xml:id="N1302D" xml:space="preserve">Equa ꝓ-
                    <lb/>
                  portiõa-
                    <lb/>
                  litas.</note>
                </s>
                <s xml:id="N12FF6" xml:space="preserve">¶ Equa aūt ꝓpor-
                  <lb/>
                tionalitas eſt duabus multitudinibus quantita-
                  <lb/>
                tum aut numerorū datis numero equalibus: et ꝓ-
                  <lb/>
                portionabilibus continuo eadem proportione: ex­
                  <lb/>
                cluſis mediis extremorum ꝓportionalis illatio.
                  <lb/>
                </s>
                <s xml:id="N13002" xml:space="preserve">Iſto modo arguendo ſicut ſe habent .1.2.4. ita .4.
                  <lb/>
                8.16. / igitur ſicut ſe habent .4. ad .16. ita .1. ad 4.</s>
              </p>
              <p xml:id="N13037">
                <s xml:id="N13038" xml:space="preserve">Poteris etiã exēplificare in aliis generibus pro-
                  <lb/>
                portionū addendo in qualibet illarū duarū mul-
                  <lb/>
                titudinū quotcun terminos volueris dūmõ ſint
                  <lb/>
                continuo ꝓportionabiles: et tot in vna multitudīe
                  <lb/>
                quot in altera. </s>
                <s xml:id="N13043" xml:space="preserve">¶ Et aduerte /  illa particula ſicut
                  <lb/>
                ſe habent que ponitur in oībus his modis arguē-
                  <lb/>
                di: denotat ſimilitudinē ſpecificã ꝓportionum.
                  <note position="left" xlink:href="note-0032-02a" xlink:label="note-0032-02" xml:id="N13072" xml:space="preserve">Denota-
                    <lb/>
                  tio illius
                    <lb/>
                  ꝑticule ſi­
                    <lb/>
                  cut ſe hꝫ:</note>
                </s>
                <s xml:id="N1304F" xml:space="preserve">Et
                  <lb/>
                intelligitur ſic ſicut ſe habēt .1.2.4. ita .3.6.12. hoc
                  <lb/>
                eſt quacun ꝓportione ꝓportionantur ſereatim
                  <lb/>
                1.2.4. / eadē ꝓportione ſpecifice ꝓportionant̄̄: 3.6.
                  <lb/>
                12. </s>
                <s xml:id="N1305A" xml:space="preserve">¶ Sed qm̄ hi ſex modi argumētandi in ꝓpor-
                  <lb/>
                tionalitatibus ſunt plurimū vſitati: et apud phi-
                  <lb/>
                loſophantes calculatores et apud primores ma-
                  <lb/>
                thematicoꝝ celebres habentur quibus magnam
                  <lb/>
                ſue doctrine partē demõſtrant: ideo nõ abs re eos
                  <lb/>
                arguendi modos in preſentiaꝝ duxi demonſtran­
                  <lb/>
                dos: qm̄ hoꝝ modoꝝ arguendi demõſtrationes ex
                  <lb/>
                precedenti capite eliciūtur facile. </s>
                <s xml:id="N1306B" xml:space="preserve">Sit igitur.</s>
              </p>
              <p xml:id="N1307E">
                <s xml:id="N1307F" xml:space="preserve">Prima concluſio. </s>
                <s xml:id="N13082" xml:space="preserve">Argumentatio a
                  <lb/>
                cõuerſa ꝓportiõalitate eſt neceſſariū argumentū.
                  <lb/>
                </s>
                <s xml:id="N13088" xml:space="preserve">Hec concluſio ſuã demonſtrationē ex tertio corre-
                  <lb/>
                lario quarte cõcluſionis precedentis capitis ſorti­
                  <lb/>
                tur: qm̄ illud correlariū principaliter oſtēdit hūc
                  <lb/>
                modū arguēdi ꝓportiõalitate cõuerſa eſſe validū</s>
              </p>
              <p xml:id="N13091">
                <s xml:id="N13092" xml:space="preserve">Secunda concluſio modus ratioci-
                  <lb/>
                nandi a ꝓportionalitate permutata ſiue cõmuta-
                  <lb/>
                ta infallibilis eſt. </s>
                <s xml:id="N13099" xml:space="preserve">Probatur hec cõcluſio manife-
                  <lb/>
                ſte ex quarta precedentis capitis. </s>
                <s xml:id="N1309E" xml:space="preserve">Idem enim hec
                  <lb/>
                et illa intendunt.</s>
              </p>
              <p xml:id="N130A3">
                <s xml:id="N130A4" xml:space="preserve">Tertia cõcluſio </s>
                <s xml:id="N130A7" xml:space="preserve">Deductio illa et mo­
                  <lb/>
                dus arguendi qui ꝓportionalitati cõiuncte īnitit̄̄
                  <lb/>
                omni exceptione eſt maior. </s>
                <s xml:id="N130AE" xml:space="preserve">Patet hec cõcluſio de-
                  <lb/>
                monſtratione euidenti ex primo correlario eiuſdē
                  <lb/>
                quarte concluſionis.</s>
              </p>
              <p xml:id="N130B5">
                <s xml:id="N130B6" xml:space="preserve">Quarta concluſio </s>
                <s xml:id="N130B9" xml:space="preserve">Forma ratiocinã­
                  <lb/>
                di a diſiūcta ꝓportiõalitate oēm exuperat inſtan-
                  <lb/>
                tiam. </s>
                <s xml:id="N130C0" xml:space="preserve">Semꝑ prauū excipio intellectū. </s>
                <s xml:id="N130C3" xml:space="preserve">Hec conclu-
                  <lb/>
                ſio patrocinante quarto correlario quarte cõclu-
                  <lb/>
                ſionis predicte manifeſta euadet.</s>
              </p>
              <p xml:id="N130CA">
                <s xml:id="N130CB" xml:space="preserve">Quinta concluſio </s>
                <s xml:id="N130CE" xml:space="preserve">Conſequentia il­
                  <lb/>
                la que ꝓportionalitas euerſa nūcupat̄̄ omne du-
                  <lb/>
                bietatis telū euertit facile: et inconcuſſa permanet
                  <lb/>
                </s>
                <s xml:id="N130D6" xml:space="preserve">Hec etiã cõcluſio quīti correlarii auxilio mõſtrat̄̄.</s>
              </p>
              <p xml:id="N130D9">
                <s xml:id="N130DA" xml:space="preserve">Sexta concluſio </s>
                <s xml:id="N130DD" xml:space="preserve">Equa argumenta­
                  <lb/>
                tio ita equitatis mediū ſureat: vt nullo inſtantie
                  <lb/>
                vicio in eã adducto ab equitatꝪ et rectitudinis tra­
                  <lb/>
                mite declinet. </s>
                <s xml:id="N130E6" xml:space="preserve">Huiꝰ concluſionis inconcuſſa equi-
                  <lb/>
                tas at īuiolata veritas clipeis et armis ſexti cor­
                  <lb/>
                relarii eiuſdē concluſionis munitur et defenſatur
                  <lb/>
                </s>
                <s xml:id="N130EE" xml:space="preserve">Et hec ad demõſtrandos predictos arguendi mo­
                  <lb/>
                dos dixiſſe ſufficiat / qm̄ illoꝝ correlarioꝝ demon-
                  <lb/>
                ſtratio harum cõcluſionum eſt euidens probatio.</s>
              </p>
              <cb chead="Capitulum quartū."/>
            </div>
            <div xml:id="N130F7" level="3" n="4" type="chapter" type-free="capitulum">
              <head xml:id="N130FC" xml:space="preserve">Capitulum quartum / in quo agitur de ex-
                <lb/>
              ceſſu cõpoſitione et diuiſione ꝓportionū.</head>
              <p xml:id="N13101">
                <s xml:id="N13102" xml:space="preserve">AD inueſtigandum paucis ex
                  <lb/>
                quibus ꝓportionibus ꝓportio aliqua
                  <lb/>
                cõponitur: in quas reſoluitur: et qua vĺ
                  <lb/>
                quibus minorē excedit: pono aliquas ſuppoſitio-
                  <lb/>
                nes quarum alique ſunt diffinitiones: et petitio-
                  <lb/>
                nes: alie vero demonſtrabuntur.</s>
              </p>
              <p xml:id="N1310F">
                <s xml:id="N13110" xml:space="preserve">Prima ſuppoſitio. </s>
                <s xml:id="N13113" xml:space="preserve">Primi termini a-
                  <lb/>
                licuius ꝓportionis ſunt illi qui in ſua ꝓportione
                  <lb/>
                ſunt minimi.
                  <note position="right" xlink:href="note-0032-03a" xlink:label="note-0032-03" xml:id="N13153" xml:space="preserve">Minimi
                    <lb/>
                  termini.</note>
                </s>
                <s xml:id="N1311F" xml:space="preserve">Minimi autē termini alicuiꝰ ꝓporti-
                  <lb/>
                onis (et loquor tam in quantitate continua quam
                  <lb/>
                diſcreta) ſunt quorū minor denominatur ab vni-
                  <lb/>
                tate: maior vero a numero vel numero cū fractiõe
                  <lb/>
                vel vnitate cū fractione. </s>
                <s xml:id="N1312A" xml:space="preserve">Hec nõ ꝓbatur / q2 diffini­
                  <lb/>
                tio eſt ſed exēplo explicatur binarius em̄ et vnitas
                  <lb/>
                ſunt primi termini ꝓportionis duple: ternarius et
                  <lb/>
                vnitas triple: quaternarius et vnitas quadruple:
                  <lb/>
                et ſic cõſequenter. </s>
                <s xml:id="N13135" xml:space="preserve">Unitas et vnitas cū medietate: et
                  <lb/>
                vnitas cū vnitate et tertia. </s>
                <s xml:id="N1313A" xml:space="preserve">Itē vnitas cū quarta et
                  <lb/>
                vnitas / et ſic cõſequenter ſunt primi termini ſuper-
                  <lb/>
                particulariū proportionum. </s>
                <s xml:id="N13141" xml:space="preserve">Unitatis .n. cum me-
                  <lb/>
                dietate ad vnitatem eſt ſexquialtera: et vnitatis
                  <lb/>
                cum tertia ad vnitatem ſexquitertia: vnitatis cum
                  <lb/>
                quarta ſexquiquarta: et ſic conſequēter. </s>
                <s xml:id="N1314A" xml:space="preserve">Et iſto mo­
                  <lb/>
                do exēplificabis in aliis generibus proportionis.</s>
              </p>
              <p xml:id="N1315B">
                <s xml:id="N1315C" xml:space="preserve">Secunda ſuppoſitio. </s>
                <s xml:id="N1315F" xml:space="preserve">Denominatio
                  <lb/>
                alicuius ꝓportionis eſt illa que ſumitur a maiori
                  <lb/>
                primoꝝ terminoꝝ talis ꝓportionis. </s>
                <s xml:id="N13166" xml:space="preserve">vt denomina­
                  <lb/>
                tio duple ſumitur a binario qui eſt maior termi-
                  <lb/>
                norū primoꝝ proportionis duple: et denominatio
                  <lb/>
                ſexquialtere ab vnitate cū dimidio.
                  <note position="right" xlink:href="note-0032-04a" xlink:label="note-0032-04" xml:id="N131E8" xml:space="preserve">1. correla­
                    <lb/>
                  rium.</note>
                </s>
                <s xml:id="N13174" xml:space="preserve">¶ Ex quo ſe-
                  <lb/>
                quitur /  ſpecies ꝓportionis multiplicis denomi­
                  <lb/>
                nãtur cõſequenter a naturali ſerie numeroꝝ. </s>
                <s xml:id="N1317B" xml:space="preserve">Ptꝫ /
                  <lb/>
                q2 maior terminus primoꝝ terminoꝝ ꝓportionis
                  <lb/>
                duple eſt binariꝰ, triple, ternariꝰ, quadruple qua­
                  <lb/>
                ternarius: et ſic conſequēter ꝓcedendo per natura­
                  <lb/>
                lē ſeriē numeroꝝ referendo numeros ad vnitatem /
                  <lb/>
                igitur ex ſecūda ſuppoſitione tales ſpecies deno-
                  <lb/>
                minantur a naturali ſerie.
                  <note position="right" xlink:href="note-0032-05a" xlink:label="note-0032-05" xml:id="N131F0" xml:space="preserve">2. correĺ.</note>
                </s>
                <s xml:id="N1318F" xml:space="preserve">¶ Sequitur ſecundo / 
                  <lb/>
                ſpecies ꝓportionis ſuperparticularis denominã­
                  <lb/>
                tur ab vnitate cū aliqua parte aliquota. </s>
                <s xml:id="N13196" xml:space="preserve">Probat̄̄ /
                  <lb/>
                q2 maior terminus primoꝝ numeroꝝ ꝓportionis
                  <lb/>
                ſexquialtere eſt vnitas cū dimidio: et ſexquitertie
                  <lb/>
                vnitas cū tertia: et ſexquiquarta cū quarta / et ſex-
                  <lb/>
                quiquinta cū quinta: et ſic conſequenter deſcendē-
                  <lb/>
                do per partes aliquotas denominatas continuo
                  <lb/>
                a naturali ſerie numeroꝝ: igitur ſpecies ꝓportio-
                  <lb/>
                nis ſuperparticularis denominantur ab vnitate
                  <lb/>
                cū parte aliquota.
                  <note position="right" xlink:href="note-0032-06a" xlink:label="note-0032-06" xml:id="N131F6" xml:space="preserve">3. correĺ.</note>
                </s>
                <s xml:id="N131AE" xml:space="preserve">¶ Sequitur tertio /  oēs ſpeci-
                  <lb/>
                es ꝓportionis ſuprapartientis denominantur ab
                  <lb/>
                vnitate cū aliquot partibus aliquotis nõ facien-
                  <lb/>
                tibus vnã. </s>
                <s xml:id="N131B7" xml:space="preserve">Probatur / q2 maior primoꝝ terminoꝝ
                  <lb/>
                ꝓportionis ſuprabipartientis tertias eſt vnitas
                  <lb/>
                cū duabus tertiis: et ſuprapartiētis quītas vni-
                  <lb/>
                tas cū duabus quintis: et ſuprabipartientis ſepti­
                  <lb/>
                mas vnitas cū duabus ſeptimis: et ſic conſequen-
                  <lb/>
                ter: diſcurrēdo per duas partes aliquotas nume-
                  <lb/>
                ri imparis. </s>
                <s xml:id="N131C6" xml:space="preserve">Item diſcurrendo per tres partes ali­
                  <lb/>
                quotas nõ facientes vnã. / per quatuor. / per quin /
                  <lb/>
                et ſic conſequenter: igitur ſpecies ꝓportionis ſu-
                  <lb/>
                prapartiētis denominãtur ab vnitate cū aliquot
                  <lb/>
                partibus aliquotis nõ facientibus vnã
                  <note position="right" xlink:href="note-0032-07a" xlink:label="note-0032-07" xml:id="N131FC" xml:space="preserve">4. correĺ.</note>
                </s>
                <s xml:id="N131D6" xml:space="preserve">¶ Sequit̄̄
                  <lb/>
                quarto /  ꝓportiones cõpoſite denominãtur a nu­
                  <lb/>
                mero cū fractione partis aliquote vel partiū ali-
                  <lb/>
                quotarū nõ facientiū vnã. </s>
                <s xml:id="N131DF" xml:space="preserve">Oſtendas hoc correla-
                  <lb/>
                riū ſicut precedentia.</s>
              </p>
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