Alvarus, Thomas, Liber de triplici motu, 1509

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            <div xml:id="N10E2D" level="3" n="5" type="chapter" type-free="capitulum">
              <p xml:id="N110EC">
                <s xml:id="N110FB" xml:space="preserve">
                  <pb chead="Prime partis" file="0014" n="14"/>
                tera: capienda eſt vna tertia pro ṗma parte: et ſex­
                  <lb/>
                quitertia. / vna quarta et ſexquiquarta vna quinta
                  <lb/>
                et ſexquiquīta vna ſexta: et ſic cõſequēter. </s>
                <s xml:id="N11107" xml:space="preserve">Probat̄̄ /
                  <lb/>
                qm̄ ad diuidēdum corpus aliqua ꝓportione: pro
                  <lb/>
                prima parte capiēdus eſt exceſſus quo numerꝰ ma­
                  <lb/>
                ior et primus talis ꝓportionis excedit numerū mi­
                  <lb/>
                norē eiuſdē ꝓportiõis: vt facile educitur ex prima
                  <lb/>
                cõcluſione adiūcta ſcḋa ſuppoſitione: ſed primus
                  <lb/>
                numerꝰ et maior ꝓportionis ſuperparticularis ex­
                  <lb/>
                cedit numeꝝ minorē ſemꝑ vna parte aliquota ſui
                  <lb/>
                denoīta a numero maiore: vt primꝰ numerꝰ et ma­
                  <lb/>
                ior ꝓportionis ſexq̇altere excedit minorē per vnã
                  <lb/>
                tertiã ſui: et primꝰ numerꝰ et maior ꝓportiõis ſex-
                  <lb/>
                quitertie excedit minorē. </s>
                <s xml:id="N11120" xml:space="preserve">per vnã quartã ſui primꝰ
                  <lb/>
                vero numerꝰ et maior ꝓportiões ſexquiquarte ex­
                  <lb/>
                cedit minorē per vnã quintaꝫ ſui: vt ptꝫ ex genera-
                  <lb/>
                tione ſpecierū ꝓportionis ſuperparticularis ca-
                  <lb/>
                pite ſecūdo huius partis: igitur diuidendo ꝓpor­
                  <lb/>
                tione ſexquialtera debet capi vna tertia ꝓ prima
                  <lb/>
                parte: et ſexquitertia vna quarta: et ſic conſequen­
                  <lb/>
                ter. </s>
                <s xml:id="N11131" xml:space="preserve">Patet igitur concluſio.
                  <note position="left" xlink:href="note-0014-01a" xlink:label="note-0014-01" xml:id="N1117D" xml:space="preserve">Correla­
                    <lb/>
                  rium.</note>
                </s>
                <s xml:id="N11139" xml:space="preserve">¶ Ex hac concluſione
                  <lb/>
                ſequitur /  diuiſo corpore per partes ꝓportiona­
                  <lb/>
                les proportiõe ſexquialtera reſiduū a prima par­
                  <lb/>
                te eſt duplum ad primū: et ſexquitertia triplū: et ſex­
                  <lb/>
                quiquarta quadruplã: et ſexquiquinta. / quintuplū
                  <lb/>
                et ſic in infinitū. </s>
                <s xml:id="N11146" xml:space="preserve">oppoſito modo ad ſpecies ꝓpor-
                  <lb/>
                tionis multiplicis incipiēdo a tripla. </s>
                <s xml:id="N1114B" xml:space="preserve">Probatur
                  <lb/>
                hoc correlariū. </s>
                <s xml:id="N11150" xml:space="preserve">qm̄ diuiſo corpore proportiõe ſex­
                  <lb/>
                quialtera prima pars eſt vna tertia. / vt ptꝫ ex pre­
                  <lb/>
                cedēti concluſione: ergo reſiduum a prima eſt due
                  <lb/>
                tertie. </s>
                <s xml:id="N11159" xml:space="preserve">Modo due tertie ſunt duplum ad vnã. </s>
                <s xml:id="N1115C" xml:space="preserve">Iteꝫ
                  <lb/>
                diuiſo corpore ꝓportiõe ſexquitertia prima pars
                  <lb/>
                corporis ē vna quarta: igit̄̄ reſiduū a prima eſt .3.
                  <lb/>
                quarte ſed triū quartarū ad vnã quartam eſt pro­
                  <lb/>
                portio tripla: igitur. </s>
                <s xml:id="N11167" xml:space="preserve">Iteꝫ diuiſo corpore ꝓportio­
                  <lb/>
                ne ſexquiquarta prima pars eſt vna quinta / vt ptꝫ
                  <lb/>
                ex prima concluſione: igit̄̄ totū reſiduū eſt .4. quin­
                  <lb/>
                te. </s>
                <s xml:id="N11170" xml:space="preserve">Modo .4. quītarum ad vnã quintam eſt pro-
                  <lb/>
                portio quadrupla et ſic de qualibet alia ꝓbabis
                  <lb/>
                </s>
                <s xml:id="N11176" xml:space="preserve">Patet iſte conſequētie ex ſecunda ſuppoſitione.</s>
              </p>
              <p xml:id="N11185">
                <s xml:id="N11186" xml:space="preserve">Quinta concluſio. </s>
                <s xml:id="N11189" xml:space="preserve">Ad diuidendum
                  <lb/>
                corpus qua placuerit ꝓportiõe ſuprapartiēti ge-
                  <lb/>
                nerentur ſpecies huiꝰ ꝓportionis ſereatim modo
                  <lb/>
                poſito in ſecundo capite huiꝰ partis: et diuidatur
                  <lb/>
                corpus in tot partes quotus eſt nūerus inferioris
                  <lb/>
                ordinis: et ex illis partibus capiantur tot pro re-
                  <lb/>
                ſiduo a prima parte ꝓportionali quotꝰ eſt nume­
                  <lb/>
                rus ſuperior: et reſiduū erit prima pars ꝓportio-
                  <lb/>
                nalis: </s>
                <s xml:id="N1119C" xml:space="preserve">Exemplū / vt cõſtituatur naturalis ſeries nu­
                  <lb/>
                meroꝝ incipiendo a ternario: et cõſtituatur inferꝰ
                  <lb/>
                ſeries omnium numerorum impariuꝫ incipiēdo a
                  <lb/>
                quinario / vt patet in figura.</s>
              </p>
              <xhtml:table xml:id="N111A5">
                <xhtml:tr xml:id="N111A6">
                  <xhtml:td xml:id="N111A7" xml:space="preserve"/>
                </xhtml:tr>
              </xhtml:table>
              <p xml:id="N111A9">
                <s xml:id="N111AA" xml:space="preserve">Tunc ſi vis diuidere aliquod corpus in ꝓportiõe
                  <lb/>
                ſuprabipartiente tertias: q2 numerus inferior in
                  <lb/>
                illa ſpecie eſt quinariꝰ diuidas totū corpꝰ ī quī
                  <lb/>
                quintas: et q2 nūerus ſuperior eſt ternariꝰ: capias
                  <lb/>
                ꝓ reſiduo a ṗma parte ꝓportionali tres quītas et
                  <lb/>
                manebūt due quīte: et ille due quīte ſunt ṗma pars
                  <lb/>
                ꝓportiõalis ꝓportiõe ſuprabipartiēte tertias. </s>
                <s xml:id="N111B9" xml:space="preserve">Et
                  <lb/>
                iſto modo in oībus aliis ſpecieꝰ operaberis. </s>
                <s xml:id="N111BE" xml:space="preserve">Et
                  <lb/>
                qm̄ in capite ſcḋo vbi generant̄̄ ſpecies huiꝰ ꝓpor­
                  <lb/>
                tionis nõ oēs generant̄̄ quãuis generent̄̄ infinite
                  <lb/>
                </s>
                <s xml:id="N111C6" xml:space="preserve">Ideo ad diuidendū corpꝰ qua volueris ꝓportiõe
                  <lb/>
                ſuprapartiēte vtaris doctrina ſecūde cõcluſionis
                  <cb chead="Capitulum quintū."/>
                </s>
                <s xml:id="N111CE" xml:space="preserve">Patet hec cõcluſio facile ex cõcluſiõe ſecūda. </s>
                <s xml:id="N111D1" xml:space="preserve">¶ Ex
                  <lb/>
                hac cõcluſione ſequit̄̄ /  in diuiſiõe corporis ṗma
                  <lb/>
                ſpecie ꝓportiõis ſuprapartientis ſignate inferiꝰ
                  <lb/>
                reſiduū a prima parte ꝓportiõali eſt ſexquialteꝝ
                  <lb/>
                ad primã: et in ſecūda ſpecie reſiduū a prima eſt ſex­
                  <lb/>
                quitertiū ad primã: et in tertia ſpecie eſt ſexquiq̈r­
                  <lb/>
                tū ad primã: et in q̈rta reſiduū a prima erit ſexqui­
                  <lb/>
                quītū ad primã: et ſic in īfinitū ꝓcedēdo ꝑ ſpecies
                  <lb/>
                ꝓportionis ſuꝑparticularis. </s>
                <s xml:id="N111E4" xml:space="preserve">Probat̄̄ hoc corre-
                  <lb/>
                lariū / qm̄ in prima ſpecie illaꝝ ſpecieꝝ generataꝝ
                  <lb/>
                in figura ꝓ reſiduo a prima parte ꝓportionali ca­
                  <lb/>
                piūtur tres quīte: et ꝓ prima parte manēt due quī­
                  <lb/>
                te / vt ptꝫ ex concluſione precedēti: ſed triū quītarū
                  <lb/>
                ad duas quītas eſt ꝓportio ſexquialtera: igr̄. </s>
                <s xml:id="N111F1" xml:space="preserve">Itē
                  <lb/>
                in ſcḋa ſpecie ꝓ reſiduo a prima parte ꝓportõali
                  <lb/>
                capiunt̄̄ quatuor ſeptime: et ꝓ prima tres ſeptime
                  <lb/>
                ſed quatuor ſeptimaꝝ ad tres ſeptimas in ꝓpor-
                  <lb/>
                tio ſexq̇tertia: igr̄. </s>
                <s xml:id="N111FC" xml:space="preserve">In tertia / vero ſpecie ꝓ reſiduo
                  <lb/>
                a prima capiūtur quī none: et pro prima reſidue
                  <lb/>
                q̈ttuor none: ſed q̇n nonaꝝ ad quattuor nonas
                  <lb/>
                eſt ꝓportio ſexquiq̈rta igit̄̄. </s>
                <s xml:id="N11205" xml:space="preserve">Et ſic ꝓbabis de qua­
                  <lb/>
                libet alia ſpecie illiꝰ figure. </s>
                <s xml:id="N1120A" xml:space="preserve">Ptꝫ igit̄̄ correlarium
                  <lb/>
                </s>
                <s xml:id="N1120E" xml:space="preserve">¶ Sed ad īueniēdã ꝓportionē reſidui a ṗma par­
                  <lb/>
                te proportiõali ad ipſam primam in reſiduis ſpe­
                  <lb/>
                ciebus conſulas ſecundam concluſionem.</s>
              </p>
              <p xml:id="N11215">
                <s xml:id="N11216" xml:space="preserve">Sexta concluſio. </s>
                <s xml:id="N11219" xml:space="preserve">Ad diuidendū cor­
                  <lb/>
                pus qua volueris ꝓportione multiplici ſuꝑparti­
                  <lb/>
                culari: generent̄̄ in nūeris ſpecies huiꝰ ꝓportiõis
                  <lb/>
                modo poſito in ſecūdo capite huiꝰ partis: et diui­
                  <lb/>
                datur corpus in tot partes quotꝰ eſt numerꝰ infe­
                  <lb/>
                rioris ordinis: et ex illis partibꝰ capiant̄̄ tot ꝓ re­
                  <lb/>
                ſiduo a prima parte ꝓportionali quotus eſt nume­
                  <lb/>
                rus ſuperior: et reſiduū erit prima pars ꝓportiõa­
                  <lb/>
                lis. </s>
                <s xml:id="N1122C" xml:space="preserve">Et eodē modo fiat diuidēdo ꝓportõe multipli­
                  <lb/>
                ci ſuprapartiēte: vt ad diuidendū corpꝰ ꝓportiõe
                  <lb/>
                dupla ſexquialtera: q2 numerꝰ maior ī illa ſpecie
                  <lb/>
                eſt quinariꝰ: diuidat̄̄ corpus in quī quītas: et q2
                  <lb/>
                numerꝰ minor eſt binariꝰ capiant̄̄ due quīte ꝓ re-
                  <lb/>
                ſiduo a prima parte ꝓportiõali: et tres quīte erūt
                  <lb/>
                ṗma pars ꝓportionalis: et tres quīte reſidui ſcḋa
                  <lb/>
                et iteꝝ tres quinte reſidui a prima et ſcḋa, tertia: et
                  <lb/>
                ſic ſine termīo. </s>
                <s xml:id="N1123F" xml:space="preserve">Itē ſi vis diuidere corpꝰ ꝓportione
                  <lb/>
                dupla ſuprabipartiēte tertias diuidas corpus in
                  <lb/>
                octo octauas: q2 nūerꝰ octonariꝰ eſt nūerꝰ maior
                  <lb/>
                illius ꝓportiõis: et capias ꝓ reſiduo a ṗma parte
                  <lb/>
                proportiõali tres octauas: et reſidue quī octaue
                  <lb/>
                erūt prima pars ꝓportiõalis: et quī octaue reſi
                  <lb/>
                dui erūt ſcḋa pars proportiõalis: et ſic cõſequēter
                  <lb/>
                </s>
                <s xml:id="N1124F" xml:space="preserve">Ptꝫ hec cõcluſio ex ſcḋa cõcluſiõe
                  <note position="right" xlink:href="note-0014-02a" xlink:label="note-0014-02" xml:id="N112F5" xml:space="preserve">Correla-
                    <lb/>
                  rium.</note>
                </s>
                <s xml:id="N11257" xml:space="preserve">¶ Ex quo ſequit̄̄ /
                  <lb/>
                 in oībus ſpeciebꝰ ꝓportiõis multiplicis ſuꝑpar­
                  <lb/>
                ticularis aut multiplicis ſuprapartiētis: et etiã in
                  <lb/>
                oībus aliis reſiduū a prima parte ꝓportionali hꝫ
                  <lb/>
                ſe ad primã partē ꝓportiõalē in ea proportõe qua
                  <lb/>
                ſe habēt nūeri ſuperiores in figuris ſuaꝝ genera-
                  <lb/>
                tionū ad nūeros ꝑ quos īferiores excedūt ſuperio­
                  <lb/>
                res: vt in proportiõe dupla ſexquialtera q2 nūerꝰ
                  <lb/>
                ſuperior eſt binariꝰ et nūerus īferior quinarius: et
                  <lb/>
                quinariꝰ excedit binariū ꝑ ternariū. </s>
                <s xml:id="N1126C" xml:space="preserve">reſiduū a pri­
                  <lb/>
                ma parte ꝓportionali in tali proportiõe ſe habet
                  <lb/>
                ad primã partē proportionalē ſicut duo ad tria et
                  <lb/>
                q2 in proportiõe dupla ſuprabipartiente tertias
                  <lb/>
                nūerus ſuperior eſt ternariꝰ: et inferior octonariꝰ
                  <lb/>
                et octonariꝰ excedit ternariū ꝑ quīariū. </s>
                <s xml:id="N11279" xml:space="preserve">ideo in ta­
                  <lb/>
                lis proportionis diuiſione reſiduū a prima parte
                  <lb/>
                proportiõali ſe hꝫ ad primã ſicut q̇nariꝰ ad terna­
                  <lb/>
                riū. </s>
                <s xml:id="N11282" xml:space="preserve">Probat̄̄ hoc correlariū ex ſecūda cõcluſione: </s>
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