Alvarus, Thomas, Liber de triplici motu, 1509

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                <s xml:id="N1384C" xml:space="preserve">
                  <pb chead="Secunde partis" file="0037" n="37"/>
                cõponitur ex duabus proportionibus ſuperpar-
                  <lb/>
                ticularibus quarum vna eſt maximi ad medium: et
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                altera medii ad minimū extremum / quod fuit pro­
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                bandum. </s>
                <s xml:id="N1386E" xml:space="preserve">Patet tamen conſequentia / quia omnis
                  <lb/>
                proportio que reperitur inter duos numeros im-
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                mediatos eſt ſuperparticularis / vt patet ex gene-
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                ratione ſuperparticulariū. </s>
                <s xml:id="N13877" xml:space="preserve">Sed tertia pars pro-
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                batur / quia duplato ſic a. et c. numero vt ſupra: iã
                  <lb/>
                a. numerus ſic duplatus excedit c. ſic duplatū per
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                dualitatem: et illa dualitas erit pars aliquota e-
                  <lb/>
                iuſdem denominationis ipſius c. ſicut antea erat
                  <lb/>
                vnitas quia adhuc manet proportio f. inter illos
                  <lb/>
                terminos: igitur adhuc maior illorum terminorū
                  <lb/>
                excedit minorem mediante eadem parte aliquota
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                minoris: diuiſa igitur illa parte aliquota a mino-
                  <lb/>
                ris que eſt dualitas in duas partes equales / puta
                  <lb/>
                in duas vnitates manifeſtum eſt /  quelibet illarū
                  <lb/>
                partium in quas diuiditur eſt pars aliquota mi-
                  <lb/>
                noris denominata a numero in duplo maiori / vt
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                conſtat: igitur numerus continens numerum mi-
                  <lb/>
                norem et talem partē aliquotam adequate ſe ha-
                  <lb/>
                bebit ad minorem numerum in proportione ſu-
                  <lb/>
                perparticulari denominata a parte aliquota que
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                denominatur a numero duplo a quo denomina-
                  <lb/>
                tur tota illa pars aliquota continens illas duas
                  <lb/>
                vnitates: et talis numerus qui videlicet cõtinet nu­
                  <lb/>
                merum minorem et medietatem illius partis ali-
                  <lb/>
                quote ſic diuiſe eſt numerus medius inter extrema
                  <lb/>
                date proportionis ſuperparticularis: igitur pro­
                  <lb/>
                portio medii termini inter terminos proportiõis
                  <lb/>
                ſuperparticularis ad minimum extremum deno-
                  <lb/>
                minatur a parte aliquota denominata a numero
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                in duplo maiore quaꝫ ſit numerus a quo denomi-
                  <lb/>
                natur pars aliquota a qua denominatur totalis
                  <lb/>
                illa proportio data ſuperparticularis. </s>
                <s xml:id="N138B2" xml:space="preserve">Conſe-
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                quētia patet: et minor probatur: quia ſemper me-
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                dius numerus inter duos excedit minorē per me-
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                dietatem exceſſus quo maior excedit minorē quia
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                alias nõ eſſet medius. </s>
                <s xml:id="N138BD" xml:space="preserve">Et ſic patet tertia pars cor­
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                relari. </s>
                <s xml:id="N138C2" xml:space="preserve">Et quarta probatur / quia ad īuento medio
                  <lb/>
                inter terminos proportionis ſuperparticularis
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                quod per ſolam vnitatem excedit numerum mino­
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                rem: et per ſolam vuitatē exceditur a maiore vt eſt
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                in propoſito: ibi reperiuntur tres numeri īmedia­
                  <lb/>
                ti in naturali ſerie numerorum / igitur proportio
                  <lb/>
                maximi eorum ad medium denominatur a parte
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                aliquota denominata a numero īmediate ſequē-
                  <lb/>
                te numerū a quo denominatur pars aliquota de-
                  <lb/>
                nominans proportionem medii numeri ad mino­
                  <lb/>
                rem / vt patet ex prima parte aſpicienti generatio-
                  <lb/>
                nem ſuperparticularium in naturali ſerie nume-
                  <lb/>
                rorum. </s>
                <s xml:id="N138DD" xml:space="preserve">Et ſic patet correlarium quadripartitum /
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                quod difficile apparet propter longitudinem ter­
                  <lb/>
                minorum quibus vtitur inprobatione.
                  <note position="left" xlink:href="note-0037-01a" xlink:label="note-0037-01" xml:id="N13A90" xml:space="preserve">Documē­
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                  tū nõ pre­
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                  tereundū</note>
                </s>
                <s xml:id="N138E9" xml:space="preserve">Et ideo de
                  <lb/>
                cetero cum voluero dicere /  aliqua proportio ſu-
                  <lb/>
                perparticularis denomīatur ab aliqua parte a-
                  <lb/>
                liquota denominata ab aliquo certo numero: di-
                  <lb/>
                cã /  talis proportio ſuperparticularis denomi-
                  <lb/>
                natur a tali numero gratia breuitatis: quia nulla
                  <lb/>
                ſuperparticularis denominatur a numero: ſed a
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                parte aliquota et vnitate: et cū dico /  denomina-
                  <lb/>
                tur a parte aliquota intelligo inadequate quod
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                ad propoſitum ſufficit.
                  <note position="left" xlink:href="note-0037-02a" xlink:label="note-0037-02" xml:id="N13A9A" xml:space="preserve">7. correĺ.</note>
                </s>
                <s xml:id="N13903" xml:space="preserve">¶ Sequitur ſeptimo /  in
                  <lb/>
                omni proportiõe ſuperparticulari capta propor­
                  <lb/>
                tione que eſt medii termini ad infimum: illa etiam
                  <lb/>
                componitur ex duabus ſuperparticularibus qua­
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                rum vna ſimiliter eſt medii termini ad infimum:
                  <lb/>
                et illa denominatur a numero quadruplo ad nu-
                  <lb/>
                merum a quo denominatur illa ſuperparticula-
                  <cb chead="Capitulum quintū."/>
                ris proportio data: vt in proportione ſexquiquar­
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                ta que eſt .20. ad .16. capta proportione que eſt in-
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                ter .18. et .16. puta medii numeri ad īfimū: illa etiã
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                cõponitur ex proportione medii termini eius pu-
                  <lb/>
                ta .17. ad .16. / et illa proportio denominatur a nu-
                  <lb/>
                mero quadruplo ad numerū a quo denominatur
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                proportio ſexquiquarta: quia ꝓportio que eſt .17.
                  <lb/>
                ad .16. denominatus a numero ſexdecimo: et pro-
                  <lb/>
                portio .20: ad .16. a numero quaternario hoc eſt a
                  <lb/>
                parte aliquota denominata ab illo puta quater­
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                nario (ſemper ſic intelligo) </s>
                <s xml:id="N13929" xml:space="preserve">Modo ſexdecimus nu­
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                merus eſt quadruplus ad quaternarium. </s>
                <s xml:id="N1392E" xml:space="preserve">Proba­
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                tur: et capio vnam proportionem ſuperparticula­
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                rem f. que ſit a. ad d. et medius numerus inter illa
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                extrema ſit b. / tunc dico /  proportio b. ad d. com-
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                ponitur ex duabus proportionibus ſuperparti-
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                cularibus quaruꝫ vna eſt medii termini ad infimū
                  <lb/>
                qui medius terminus inter b. et d. ſit c. et illa puta
                  <lb/>
                c. ad d. denominatur a numero quadruplo ad nu-
                  <lb/>
                merū a quo denominatur proportio a. ad .d. </s>
                <s xml:id="N13941" xml:space="preserve">Pri­
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                ma pars videlicet /  ꝓportio que eſt b. ad d. com-
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                ponitur ex duabus ſuperparticularibꝰ .etc̈. / patet
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                ex īmediate precedenti: et ſecunda probatur / quia
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                proportio b. ad d. denominatur a numero duplo
                  <lb/>
                ad numerum a quo denominatur f. ꝓportio a. ad
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                d. / vt patet ex precedenti correlario: et proportio c.
                  <lb/>
                ad d. eadē ratione denominatur a numero duplo
                  <lb/>
                ad numerū a quo denominatur proportio b. ad d /
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                vt patet ex eodem correlario: igitur proportio c.
                  <lb/>
                ad d. denomīatur a numero quadruplo ad nume­
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                merū a quo denominatur ꝓportio f.a. ad d. / quod
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                fuit probandū. </s>
                <s xml:id="N1395C" xml:space="preserve">Patet hec conſequentia: quia nu-
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                merus duplus ad duplū alicuiꝰ certi dati eſt qua-
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                druplus ad illum certum datum / vt conſtat: ſed nu­
                  <lb/>
                merus a quo denomīatur proportio c. ad d. eſt du­
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                plus ad numerum a quo denominatur proportio
                  <lb/>
                b. ad d. et ille iterum eſt duplus ad numeruꝫ a quo
                  <lb/>
                denominatur proportio f.a. ad d. / igitur numerus
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                a quo denominatur proportio c. ad d. eſt quadru-
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                plus ad numerum a quo denominatnr proportio
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                f. que eſt a. ad d. / quod fuit probandū.
                  <note position="right" xlink:href="note-0037-03a" xlink:label="note-0037-03" xml:id="N13AA0" xml:space="preserve">8. correĺ.</note>
                </s>
                <s xml:id="N13976" xml:space="preserve">¶ Sequitur
                  <lb/>
                octauo /  quacun proportione ſuperparticula-
                  <lb/>
                ri data denomīata ab aliquo certo numero: oīs
                  <lb/>
                proportio ſuperparticularis denominata a ma-
                  <lb/>
                iori numero vſ ad duplū incluſiue eſt maior quã
                  <lb/>
                medietas illius proportionis ſuperparticularis
                  <lb/>
                date: vt data proportione ſexquiquarta oīs pro-
                  <lb/>
                portio ſuperparticularis denominata ab olique
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                numero a quaternario vſ ad octonarium inclu-
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                ſiue qui eſt numerus duplus ad quaternarium eſt
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                maior quam ſubdupla ad ſexquiquartã et ſic ſex-
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                quiquarta, ſexquiſexta, ſexquiſeptima, ſexq̇octa-
                  <lb/>
                ua, eſt maior quam ſubdupla ad ſexquiquartam.
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                </s>
                <s xml:id="N13992" xml:space="preserve">Probatur / quoniã quacun tali ſuperparticula-
                  <lb/>
                ri data ab aliquo numero denominata: propor-
                  <lb/>
                tio ſuperparticularis denominata a numero in
                  <lb/>
                duplo maiore eſt maior quam ſubdupla ad illam
                  <lb/>
                quia talis eſt medii termini ad infimū / vt patet ex
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                quinto et ſexto correlario cõiunctis: igitur omnis
                  <lb/>
                ꝓportio ſuperparticularis denominata a nume-
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                ro minori quã duplo ad numerū a quo denomina­
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                tur data ꝓportio ſuꝑparticularis eſt maior quã
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                ſubdupla ad illam datã ſuperparticularē. </s>
                <s xml:id="N139A7" xml:space="preserve">Patet
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                hec cõſequentia per hoc /  oīs ſuperparticularis
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                que denomīatur a minori numero eſt maior: quia
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                talis denomīatur a maiori parte aliquota: et hoc
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                auxiliante loco a maiori: et per conſequens pro-
                  <lb/>
                portione ſuperparticulari data denominata ab
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                aliquo certo nūero: oīs ꝓportio ſuꝑparticularis </s>
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