Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N10B6A" level="3" n="4" type="chapter" type-free="capitulum">
              <p xml:id="N10D7E">
                <s xml:id="N10DFB" xml:space="preserve">
                  <pb chead="Prime partis" file="0012" n="12"/>
                nitū magnꝰ erit exceſſus quo quantitas maior ex­
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                cedet minorē. </s>
                <s xml:id="N10E05" xml:space="preserve">igitur in infinitū magna erit ꝓpor-
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                tio quãtitatis maior ad minorē: et per cõſequens
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                illarū infinitarū proportionū in infinitū magna
                  <lb/>
                erit aliqua: quod fuit probandū. </s>
                <s xml:id="N10E0E" xml:space="preserve">Et ſic patet con-
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                cluſio. </s>
                <s xml:id="N10E13" xml:space="preserve">¶ Simile correlariū: correlario ṗme cõclu-
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                ſiõis: hic poteris inferre de gñatione huiuſmodi
                  <lb/>
                proportionū irrationaliū. </s>
                <s xml:id="N10E1A" xml:space="preserve">¶ Plures adieciſſem
                  <lb/>
                cõcluſiones et correlaria: niſi obſtaret hanc mate­
                  <lb/>
                riã ex ſecunda parte in vniuerſum dependere. </s>
                <s xml:id="N10E21" xml:space="preserve">Nec
                  <lb/>
                mirari oportet: ſi plurimū in his duobus capitibꝰ
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                cõtra morē et ordinē mathematicū: ſequētibꝰ vſus
                  <lb/>
                fuerim. </s>
                <s xml:id="N10E2A" xml:space="preserve">Non em̄ potuit hec materia alio mõ īduci</s>
              </p>
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            <div xml:id="N10E2D" level="3" n="5" type="chapter" type-free="capitulum">
              <head xml:id="N10E32" xml:space="preserve">Capitulū quintū / in quo agit̄̄ de diuiſione
                <lb/>
              corporis in partes proportionales qua pro­
                <lb/>
              portione rationali quis voluerit.</head>
              <p xml:id="N10E39">
                <s xml:id="N10E3A" xml:space="preserve">QUoniam plerū in materia
                  <lb/>
                triplicis motus occurūt pleri caſus:
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                in quibus oportet vti multiplici ſpecie
                  <lb/>
                diuiſionis corporis in partes ſuas proportiona­
                  <lb/>
                les variis et diuerſis ꝓportionibus rationalibus
                  <lb/>
                ideo ad vniuerſalē methodū inueniendam ſit.</s>
              </p>
              <p xml:id="N10E47">
                <s xml:id="N10E48" xml:space="preserve">Prīa ſuppõ. </s>
                <s xml:id="N10E4B" xml:space="preserve">Nõ oēs ꝑtes alicuiꝰ cor­
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                poris ī q̈s idē corpꝰ diuidit̄̄ ↄ̨tinuo ſe hñtes ī eadē
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                ꝓportiõe: gr̄a exēpli a. ſūt oēs ꝑtes ꝓportionales
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                eiuſdē corꝑis eadē ꝓportiõe a. </s>
                <s xml:id="N10E54" xml:space="preserve">Probat̄̄ / q2 poſſibi­
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                le eſt /  vna medietas alicuiꝰ corꝑis diuidat̄̄ in oēs
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                partes ſuas ꝓportione tripla: et omēs ille partes
                  <lb/>
                ſunt partes illiꝰ corporis totalis. </s>
                <s xml:id="N10E5D" xml:space="preserve">in q̈s idē corpꝰ
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                diuidit̄̄ hñtes ſe cõtinuo in ꝓportiõe tripla: 2. et tñ
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                nõ ſunt oēs partes ꝓportionales illius corporis
                  <lb/>
                proportione tripla. </s>
                <s xml:id="N10E66" xml:space="preserve">Et capio in ſuppoſitiõe ly oēs
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                collectiue in primo loco et in ſecundo.</s>
              </p>
              <p xml:id="N10E6B">
                <s xml:id="N10E6C" xml:space="preserve">Secūda ſuppoſitio. </s>
                <s xml:id="N10E6F" xml:space="preserve">Oēs partes ali­
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                cuius corporis innuite continue ſe habētes aliq̈
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                ꝓportione: puta a. et abſoluentes totū corpꝰ: ſunt
                  <lb/>
                oēs partes ꝓportionales eiuſdē corporis propor­
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                tione a. </s>
                <s xml:id="N10E7A" xml:space="preserve">Et volo dicere /  ſi aliquod corpꝰ diuidat̄̄
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                in infinitas partes continuo ſe habentes in ꝓpor­
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                tione a. et abſoluētes totū corpus: ille ſimul ſunt
                  <lb/>
                oēs partes proportionales proportione a. </s>
                <s xml:id="N10E83" xml:space="preserve">Patꝫ
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                hec ſuppoſitio: q2 ſic diuidere corpus eſt diuidere
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                ipſū in oēs partes ꝓportionales proportione a.
                  <lb/>
                </s>
                <s xml:id="N10E8B" xml:space="preserve">Patet hoc ex deſcriptione termini.</s>
              </p>
              <p xml:id="N10E8E">
                <s xml:id="N10E8F" xml:space="preserve">Tertia ſuppoſitio. </s>
                <s xml:id="N10E92" xml:space="preserve">Quãdocun ali­
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                qua cõtinuo ꝓportionãtur aliqua ꝓportione geo­
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                metrica: qualis eſt ꝓportio inter proportionata:
                  <lb/>
                talis eſt inter ſuas differētias ſiue exceſſeus: quod
                  <lb/>
                idem eſt: vt q2 .3. ad .4. ſe habet in ꝓportiõe dupla
                  <lb/>
                et ſimiliter .4. ad 2. / et cõtinuo proportionant̄̄ eadē
                  <lb/>
                proportione: ideo differentia ſiue exceſſus inter .8
                  <lb/>
                et .4. ſe habet ad differãtiã ſiue exceſſum inter .4. et
                  <lb/>
                2. in proportiõe dupla. </s>
                <s xml:id="N10EA5" xml:space="preserve">Patet hec ſuppoſitio ex
                  <lb/>
                quīta proprietate proportionalitatis ſiue medie­
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                tatis geometrice ex ſecūda parte capitulo ſecūdo</s>
              </p>
              <p xml:id="N10EAC">
                <s xml:id="N10EAD" xml:space="preserve">Quarta ſuppoſitio. </s>
                <s xml:id="N10EB0" xml:space="preserve">Si aliquod cor­
                  <lb/>
                pus diuidatur in infinitas partes: et deperdendo
                  <lb/>
                primã illarū perdit aliquã ꝓportionē puta a. / hoc
                  <lb/>
                eſt efficitur in a. ꝓportione minꝰ: et ꝑdendo ſcḋam
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                poſt primã iterum efficitur in a. minus: et ꝑdendo
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                tertiam poſt ſecūdã iterum efficitur in a. minus. </s>
                <s xml:id="N10EBD" xml:space="preserve">et
                  <lb/>
                ſic conſequenter ille partes ſunt oēs partes ꝓpor­
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                tionales illius corporis ꝓportione a. / ſi vero ꝑden­
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                do primã illarū non perdit vnam proportionē a. /
                  <cb chead="Capitulum quintū."/>
                et ꝑdendo ſecundã poſt primã: vnã alteram, ꝑden-
                  <lb/>
                do tertiã poſt ſecundã vnã alteram ꝓportionē a. /
                  <lb/>
                et ſic cõſequenter: tales partes nõ ſunt oēs partes
                  <lb/>
                ꝓportionales talis corporis ꝓportione a. </s>
                <s xml:id="N10ECF" xml:space="preserve">Pro-
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                batur prima pars / q2 ſi nõ: detur oppoſitū: videli­
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                cet /  aliquod corpus diuiditur in aliquas partes
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                iufinitas: et ꝑdēdo primã illarum ꝑdit ꝓportionē
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                a. etc̈. et tamen nõ ſunt ille oēs partes ꝓportiona-
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                les illius corporis ꝓportiõe a. et ſic tale corpus b. /
                  <lb/>
                et arguitur ſic / b. eſt diuiſum in infinitas partes: et
                  <lb/>
                ꝑdendo primã illarū in prima parte ꝓportionali
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                hore exempli gratia: in fine illius partis eſt in a.
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                ꝓportiõe minꝰ: et ꝑdendo ſecundã partē in ſecūda
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                parte ꝓportionali tēporis: iterum efficitur in fine
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                eiuſdem partis in a. proportione minꝰ quaꝫ erat
                  <lb/>
                in principio eiuſdē partis: et in tertia parte ꝓpor­
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                tionali ꝑdēdo terntiã ip̄m efficitur minꝰ / quã erat
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                in principio eiuſdē ꝑtis in a. ꝓportione: et ſic con­
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                ſequēter. </s>
                <s xml:id="N10EF0" xml:space="preserve">igitur in partibus ꝓportionabilibꝰ illiꝰ
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                hore ſunt infinita corpora cõtinuo ſe habentia in
                  <lb/>
                ꝓportione a. </s>
                <s xml:id="N10EF7" xml:space="preserve">Patet / q2 corpus qḋ eſt in principio
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                p̄me partis ꝓportionalis: ſe habet in ꝓportione
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                a. ad illud quod eſt in prīcipio ſecunde et illud qḋ
                  <lb/>
                eſt in p̄ncipio ſecunde ſe habet in ꝓportione a. ad
                  <lb/>
                illud quod eſt in principio tertie: et ſic cõſequēter /
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                igitur illa infinta corpora continuo ſe habet in
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                ꝓportiõe a. / et ex cõſequēti ſequit̄̄ /  exceſſus inter
                  <lb/>
                illa corpora cõtinuo ſe habēt in ꝓportiõe a. / puta
                  <lb/>
                exceſſus quo corpus in p̄ncipio ṗme partis ꝓpor­
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                tionalis excedit corpus in p̄ncipio ſecunde: ſe ha­
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                bet in ꝓportione a. / ad exceſſum quo corpus in p̄n­
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                cipio ſecūde excedit corpus in p̄ncipio tertie: et ſic
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                cõſequēter. </s>
                <s xml:id="N10F12" xml:space="preserve">Patet hec cõſequētia ex p̄cedenti ſup­
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                poſitione: et illi exceſſus ſunt ille partes que deper­
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                dūtur in partibus ꝓportionalibus tēporis: ergo
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                ille ꝑtes que deꝑduntur in illis partibus propor-
                  <lb/>
                tionalibus tēporis ſe habent cõtinuo in ꝓportõe
                  <lb/>
                a. </s>
                <s xml:id="N10F1F" xml:space="preserve">Conſequētia patet: et ꝓbatur antecedens: quia
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                corpus in principio p̄me partis ꝓportionalis tē-
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                poris: exedit corpus in principio ſecunde ꝑ illud
                  <lb/>
                quod deꝑdit in ip̄a p̄ma parte ꝓportionali tēpo-
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                ris: et illud eſt p̄ma illarum partiū in quas diuidi­
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                tur corpus ex caſu: igitur aſſumptum verum </s>
                <s xml:id="N10F2C" xml:space="preserve">Qm̄
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                ſic ꝓbabis de quocū alio exceſſu. </s>
                <s xml:id="N10F31" xml:space="preserve">et vltra ille par­
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                tes in quas diuiditur illud corpus b. ſunt infinite
                  <lb/>
                cõtinuo ſe habentes in ꝓportione a. / et abſoluūt to­
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                tum corpus: igitur ille ſunt oēs partes ꝓportiona­
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                les illius corporis ꝓportione a. / quod fuit negatū
                  <lb/>
                </s>
                <s xml:id="N10F3D" xml:space="preserve">Patet hec conſequentia ex ſecunda ſuppoſitione.
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                </s>
                <s xml:id="N10F41" xml:space="preserve">Quod vero ille partes abſoluant totum corpus
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                patet / quia per deperditionem illarū perditur to­
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                tum corpus ad nõ quantum: cum deperdat infini­
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                tam latitudinem proportionis: vt conſtat: igitur.
                  <lb/>
                </s>
                <s xml:id="N10F4B" xml:space="preserve">Secūda pars patet facile / quia bene ſequitur de-
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                perdendo illas partes continuo: tale corpus non
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                continuo efficitur minus in proportione a. / ergo
                  <lb/>
                ſequitur /  non ſunt ibi in tali diminutione infini­
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                ta corpora continuo ſe habentia in proportione
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                a. modo ſuperius expoſito: ergo ſequitur /  exceſ­
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                ſus illorum corporum non continuo ſe habent in
                  <lb/>
                proportione a. </s>
                <s xml:id="N10F5C" xml:space="preserve">Patet conſequentia ex tertia ſup­
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                poſitione: et illi exceſſus ſunt partes in quas diui­
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                debatur ipſum corpus b. / igitur ipſe non ſunt par­
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                tes proportionales corporis b. proportione a. / et
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                per conſequens de primo ad vltimum ſequitur il­
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                la ſecunda pars ſuppoſitionis.</s>
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