Alvarus, Thomas, Liber de triplici motu, 1509

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                  <pb chead="Prime partis" file="0015" n="15"/>
                qm̄ iuxta illam cõcluſionē reſiduū a prima parte
                  <lb/>
                ꝓportionali quauis ꝓportione rationali debet ſe
                  <lb/>
                habere vt numerꝰ minor talis ꝓportionis: et ꝑ cõ­
                  <lb/>
                ſequēs manebit ꝓ prima parte ꝓportiõali nume­
                  <lb/>
                rus ille quo numerꝰ maior talis ꝓportionis exce-
                  <lb/>
                dit minorē. </s>
                <s xml:id="N11294" xml:space="preserve">Patet hec cõſequētia / q2 ſemꝑ corpus
                  <lb/>
                debet diuidi in tot partes quotus eſt numerꝰ ma-
                  <lb/>
                ior et primus ꝓportiõis qua debet fieri diuiſio: vt
                  <lb/>
                patet ex ſecūda cõcluſione: et pro reſiduo a prima
                  <lb/>
                debent capi tot partes ex illis quotus eſt numerꝰ
                  <lb/>
                minor vt dictum eſt. </s>
                <s xml:id="N112A1" xml:space="preserve">igitur relique partes remanē­
                  <lb/>
                tes erunt prima pars. </s>
                <s xml:id="N112A6" xml:space="preserve">Patet cõſequētia ex prima
                  <lb/>
                ſuppoſitione: et ille partes remanentes ſunt nume­
                  <lb/>
                rus quo numerus maior excedit minorē, vt patet:
                  <lb/>
                igitur prima pars ꝓportionalis eſt numerus quo
                  <lb/>
                maior numerꝰ et primꝰ proportionis qua ſit diui­
                  <lb/>
                ſio excedit minorē. </s>
                <s xml:id="N112B3" xml:space="preserve">Habet ſe / igitur totū reſiduū a
                  <lb/>
                prima parte proportionali ad primã partē pro-
                  <lb/>
                portionalē in ea proportione qua numerꝰ minor
                  <lb/>
                et primus talis proportionis ſe habet ad numerū
                  <lb/>
                quo maior et primus eiuſdem proportiõis excedit
                  <lb/>
                minorem. </s>
                <s xml:id="N112C0" xml:space="preserve">quod fuit probandum </s>
                <s xml:id="N112C3" xml:space="preserve">¶ Ad habendam
                  <lb/>
                autē praxim huius correlarii in cõpoſitis propor­
                  <lb/>
                tionibus conſtituētur alique figure: quibus facile
                  <lb/>
                iudicabitur in qua proportiõe ſe habet reſiduū a
                  <lb/>
                prima parte ꝓportionali ad primã partē ꝓpor-
                  <lb/>
                tionalē. </s>
                <s xml:id="N112D0" xml:space="preserve">Ad quod facile inſpiciendū in ꝓportioni­
                  <lb/>
                bus duplis ſuperparticularibus conſtituatur na­
                  <lb/>
                turalis ſeries numeroꝝ incipiēdo a binario in īfe­
                  <lb/>
                riori linea: et in ſuperiori linea conſtituatur natu­
                  <lb/>
                ralis ordo numerorū incipiendo a ternario: tunc
                  <lb/>
                referendo primum inferioris ordinis. </s>
                <s xml:id="N112DD" xml:space="preserve">primo ſu-
                  <lb/>
                periois: habebis in qua ꝓportione ſe habet reſi-
                  <lb/>
                duū a prima parte proportiõali ad primã diuidē­
                  <lb/>
                do corpus prima ſpecie ꝓportionis duple ſuper-
                  <lb/>
                particularis: et referendo ſecundū inferioris ordi­
                  <lb/>
                nis ſecundo ſuperioris habebis illud idem in ſe-
                  <lb/>
                cunda ſpecie ꝓportionis duple ſuperparticula­
                  <lb/>
                ris. </s>
                <s xml:id="N112EE" xml:space="preserve">et ſic conſequenter vt patet in figura.</s>
              </p>
              <xhtml:table xml:id="N112FD">
                <xhtml:tr xml:id="N112FE">
                  <xhtml:td xml:id="N112FF" xml:space="preserve"/>
                </xhtml:tr>
              </xhtml:table>
              <p xml:id="N11301">
                <s xml:id="N11302" xml:space="preserve">Sed ad praxim huiꝰ negocii in ſpeciebus ꝓporti­
                  <lb/>
                onis triple ſuꝑparticularis cõſtituatur in inferio­
                  <lb/>
                ri ſerie naturalis ordo numerorū incipiendo a bi­
                  <lb/>
                nario: et in ſuperiori conſtituãtur oēs numeri īpa­
                  <lb/>
                res incipiendo a quinario: et tunc referēdo primū
                  <lb/>
                inferioris ordinis primo ſuperioris: et ſecundū in­
                  <lb/>
                ferioris ſecūdo ſuperioris: et tertiū inferioris ter-
                  <lb/>
                tio ſuperioris: et ſic conſequenter. </s>
                <s xml:id="N11313" xml:space="preserve">cõſpicies in qua
                  <lb/>
                ꝓportione ſe habet reſiduum a prima parte pro­
                  <lb/>
                portionali ad primã diuiſione corporis facto pro­
                  <lb/>
                portione tripla ſuperparticulari: vt ptꝫ in figura</s>
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              <p xml:id="N11320">
                <s xml:id="N11321" xml:space="preserve">Ad praticandū autē ita in ſpeciebus quadruple
                  <lb/>
                ſuꝑparticularis quintuple ſuꝑparticularis .etc̈. / cõ­
                  <lb/>
                ſtituatur naturalis ſeries numerorū incipiendo a
                  <lb/>
                binario in linea inferiori: et in ſuperiori oēs nume­
                  <lb/>
                ros excedentes ſe continuo ternario incipiendo a
                  <lb/>
                ſeptenario: et ſic habebis quod queris in ſpeciebꝰ
                  <lb/>
                ꝓportionis quadruple ſuꝑparticularis </s>
                <s xml:id="N11330" xml:space="preserve">Ad quod
                  <lb/>
                inueniēdū in ſpeciebus ꝓportionis quītuple ſuꝑ­
                  <lb/>
                particularis cõſtituas in ſuperiori ordine oēs nu­
                  <lb/>
                meros excedentes ſe quaternario incipiendo a nu­
                  <lb/>
                mero nouenario: et in ſpecie ſequeuti coſtituas in
                  <lb/>
                ſuperiori ordine oēs numeros excedentes ſe qui­
                  <cb chead="Capitulum ſextū."/>
                nario incipiendo a numero vndenario: et ſic conſe­
                  <lb/>
                quenter in aliis ſpeciebus operaberis </s>
                <s xml:id="N11342" xml:space="preserve">Patet hoc
                  <lb/>
                in figuris ſequentibus.</s>
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              <xhtml:table xml:id="N11347">
                <xhtml:tr xml:id="N11348">
                  <xhtml:td xml:id="N11349" xml:space="preserve"/>
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              <p xml:id="N1134B">
                <s xml:id="N1134C" xml:space="preserve">¶ Sed ad exercitiū huiꝰ vltimi correlarii in ſpecie­
                  <lb/>
                bus multipliciū ſuprapartientiū quedã etiaꝫ con-
                  <lb/>
                ſtituentur figuere. </s>
                <s xml:id="N11353" xml:space="preserve">Unde ac facile īueniendã ꝓpor­
                  <lb/>
                tionē reſidui a prima parte ꝓportionali ad ipſaꝫ
                  <lb/>
                primã in ſpeciebus ꝓportionis duple ſupraparti­
                  <lb/>
                entis cõſtituatur naturalis ſeries incipiēdo a ter­
                  <lb/>
                nario inferiori linea: in ſuperiori vero cõſtituan-
                  <lb/>
                tur oēs numeri īpares incipiēdo a quinario: et tūc
                  <lb/>
                referēdo primū inferioris ordinis primo ſuperio­
                  <lb/>
                ris: et ſcḋm ſcḋo: et tertiū tertio id quod queris fa-
                  <lb/>
                cile reperies / vt patet in figura ſequenti.</s>
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                <xhtml:tr xml:id="N11367">
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              <p xml:id="N1136A">
                <s xml:id="N1136B" xml:space="preserve">¶ Ad īueniendã autē proportionē reſidui a prima
                  <lb/>
                parte ꝓportionali ad ipſam primã diuiſione cor­
                  <lb/>
                poris facta ꝓportione tripla ſuprapartiente con­
                  <lb/>
                ſtituatur ſupra naturalē ſeriē numeroꝝ incipiēdo
                  <lb/>
                a ternario vna ſeries omnium numerorum conti-
                  <lb/>
                nuo excedentium ſe ternario incipiendo ab octo-
                  <lb/>
                nario numero: vt patet in figura.</s>
              </p>
              <xhtml:table xml:id="N1137A">
                <xhtml:tr xml:id="N1137B">
                  <xhtml:td xml:id="N1137C" xml:space="preserve"/>
                </xhtml:tr>
              </xhtml:table>
              <p xml:id="N1137E">
                <s xml:id="N1137F" xml:space="preserve">¶ Ad īueniendū autē ꝓpoſitū in ſpeciebus ꝓpor-
                  <lb/>
                tionis quadruple ſuprapartiētis ſupra naturalē
                  <lb/>
                ſeriē numeroꝝ incipiendo a ternario conſtituatur
                  <lb/>
                ſeries numeroꝝ ↄ̨tinuo excedentiū ſe quaternario
                  <lb/>
                incipiendo ab vndeuario: et ſic cõſequenter ſupra
                  <lb/>
                eandē naturalē ſeriē numeroꝝ incipiendo a terna­
                  <lb/>
                rio cõſtituatur ſeries numeroꝝ cõtinuo exedentiū
                  <lb/>
                ſe numero quinario īcipiēdo a numero quarto de­
                  <lb/>
                cimo: et ſic cõſequenter operaberis in aliis. </s>
                <s xml:id="N11392" xml:space="preserve">Et hec
                  <lb/>
                de diuiſione corpoꝝ ꝓportione rationali.</s>
              </p>
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            <div xml:id="N11397" level="3" n="6" type="chapter" type-free="capitulum">
              <head xml:id="N1139C" xml:space="preserve">Capitulū ſextū / ī quo datur modus di­
                <lb/>
              uidendi corpus in partes proportiona-
                <lb/>
              les proportione irrationali.</head>
              <p xml:id="N113A3">
                <s xml:id="N113A4" xml:space="preserve">QUemadmodū quodlibet cor-
                  <lb/>
                pus diuidi poteſt ꝓportione rationali
                  <lb/>
                infinitiſ ſpeciebus eius / vt caput prece­
                  <lb/>
                dens oſtendit: ita etiã ꝓportione irrationali infi-
                  <lb/>
                nitiſ ſpeciebus eiꝰ quodlibet corpꝰ diuidi poteſt
                  <lb/>
                </s>
                <s xml:id="N113B0" xml:space="preserve">Pro cuius diuiſionis noticia ſit</s>
              </p>
              <p xml:id="N113B3">
                <s xml:id="N113B4" xml:space="preserve">Prima concluſio </s>
                <s xml:id="N113B7" xml:space="preserve">Quodlibet corpus
                  <lb/>
                diuiſū aliqua ꝓportione irrationali ſe debet ha­
                  <lb/>
                bere ad aggregatū ex oībus partibus ꝓportiona­
                  <lb/>
                bilibus tali ꝓportione ſequētibus primam in ea
                  <lb/>
                proportione qua totum diuidatur. </s>
                <s xml:id="N113C2" xml:space="preserve">Hec concluſio
                  <lb/>
                claram et euidentem ex prima precedentis capitis
                  <lb/>
                demonſtrationem ſortitur.</s>
              </p>
              <p xml:id="N113C9">
                <s xml:id="N113CA" xml:space="preserve">Secunda cõcluſio. </s>
                <s xml:id="N113CD" xml:space="preserve">Ad diuidendum
                  <lb/>
                corpus infinitis ꝓportionibꝰ irrationabilibꝰ mi­
                  <lb/>
                noribus dupla: vt puta ꝓportione diametri ad co­
                  <lb/>
                ſtam: aggregati ex medietate exceſſus quo diame­
                  <lb/>
                ter excedit coſtã et ipſa coſta ipſammet coſtam: </s>
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